Rigidity of action of compact quantum groups on compact, connected ...

Rigidity of action of compact quantum groups on compact, connected manifolds

arXiv:1309.1294v2 [math.OA] 26 Mar 2014

Biswarup Das1 , Debashish Goswami2 , Soumalya Joardar

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Indian Statistical Institute 203, B. T. Road, Kolkata 700108 Email: [email protected] Abstract Suppose that a compact quantum group Q acts faithfully on a smooth, compact, connected manifold M such that the action α is smooth, i.e. it leaves C ∞ (M ) invariant and the linear span of α(C ∞ (M ))(1 ⊗ Q) is Frechet-dense in C ∞ (M, Q). We prove that Q must be commutative as a C ∗ algebra i.e. Q ∼ = C(G) for some compact group G acting smoothly on M. As a consequence, it follows that there is no genuine quantum isometry (in the sense of [15]) of a classical compact connected Riemannian manifold, and hence the quantum isometry group of any noncommutative Riemannian manifold obtained by some cocycle twisting of a compact connected Riemannian manifold is isomorphic with similar cocycle twisted quantum version of the classical isometry group of the manifold. We also obtain similar rigidity results for more general quantum group symmetries of Rieffel-deformation of compact connected manifolds.

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Introduction

It is a very important and interesting problem in the theory of quantum groups and noncommutative geometry to study ‘quantum symmetries’ of various classical and quantum structures. Indeed, symmetries of physical systems (classical or quantum) were conventionally modelled by group actions, and after the advent of quantum groups, group symmetries were naturally generalized to symmetries given by quantum group action. In this context, it is natural to think of quantum automorphism or the full quantum symmetry groups of various mathematical and physical structures. The underlying basic principle of defining a quantum automorphism group of a given mathematical structure consists of two steps : first, to identify (if possible) the group of automorphisms of the structure as a universal object in a suitable category, and then, try to look for the universal object in a similar but bigger category by replacing groups by quantum groups of appropriate type. Initiated by S. Wang who defined and studied quantum permutation groups of finite sets and quantum automorphism groups of finite dimensional algebras, such questions were taken up by a number of mathematicians including Banica, Bichon (see, e.g. [3], [4], [40]), and 1 Acknowledges

support from UKIERI supported by Swarnajayanti Fellowship from D.S.T. (Govt. of India) and also acknowledges the Fields Institute, Toronto for providing hospotality for a brief stay when a small part of this work was done 3 Acknowledges support from CSIR 2 Partially

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more recently in the framework of Connes’ noncommutative geometry ([12]) by Goswami, Bhowmick, Skalski, Banica and others who have extensively studied the quantum group of isometries (or quantum isometry group) defined in [15] (see also [11], [9], [7] etc.). In this context, it is important to compute such quantum isometry groups for classical (compact) Riemannian manifolds. This will also allow one to compute quantum isometry groups of the noncommutative manifolds obtained by Rieffel-type deformation of classical manifolds by the techniques developed in [8], and more generally, for examples obtained by cocycle-twists to be elaborated in a forthcoming paper ([18]). we are also working to accomodate the examples obtained by deformation by Drinfeld twists as in [27]. However, it was rather amazing to see that for all the connected classical manifolds including the spheres and the tori (with the usual Riemannian metrics) for which the quantum isometry groups were computed so far, the quantum isometry groups turned out to be the same as the classical isometry groups. In other words, there is no genuine (i.e. noncommutative as a C ∗ algebra) compact quantum group which can act isometrically on such manifolds. It may be mentioned here that it is easy to have faithful isometric action of genuine compact quantum group on disconnected Riemannian manifolds with at least four components. However, no examples of even faithful continuous action by genuine compact quantum groups on C(X) with X being connected compact space were known until recently, when H. Huang ([19]) constructed examples of such action on topological spaces which are typically obtained by topological connected sums of copies of some given compact metric space. But none of the examples constructed by Huang are smooth manifolds. In fact, his construction would fail if topological connected sum is replaced by a smooth gluing of copies of a given Riemannian manifold. On the other hand, it follows from the work of Banica et al ([5]) that most of known compact quantum groups, including the quantum permutation groups of Wang, can never act faithfully and isometrically on a connected compact Riemannian manifold. All these led the second author of the present paper to make the following conjecture in [16], where he also gave some supporting evidence to this conjecture, by proving non-existence of ‘linear’ (see [16] for the precise definition) action of any genuine compact quantum group on a large class of classical connected manifolds which are homogeneous spaces of semisimple compact connected Lie groups. Conjecture I: It is not possible to have smooth faithful actions of genuine compact quantum groups on C(M ) when M is a compact connected smooth manifold. In this paper, we prove this conjecture. More precisely, we have: Theorem I (Theorem 11.9 of Section 11 ) Suppose that a CQG Q acts faithfully on a smooth, compact connected manifold M such that the action α is smooth, i.e. it leaves C ∞ (M ) invariant and the linear span of α(C ∞ (M ))(1 ⊗ Q) is Frechet-dense in C ∞ (M, Q), We prove that Q must be commutative as a C ∗ algebra i.e. Q ∼ = C(G) for some compact group G acting smoothly on M .

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For classical group action, smoothness is a topological condition (continuity of the action with respect to the natural Frechet topology of smooth function algebra) which automatically implies an algebraic condition, namely, the differantial of the action gives a smooth representation of the group on the bimodule of one-forms. However, this may not hold for quantum group actions; we do have a counter-example at least in the algebraic set-up (see Example 2, Section 13). and we believe that a counter-example will exist in the analytic set-up as well for actions of locally compact quantum groups on possibly noncompact manifolds. Fortunately, for smooth CQG actions as in Theorem I we always get a well-defined representation on the bimodule of one-forms. It may be noted that isometric actions always satisfy the smoothness hypothesis of the above theorem, hence we get the following: Theorem II (Corollary 11.8 of Section 11): For any compact, connected Riemannian manifold M , the quantum isometry group is classical, i.e. same as C(ISO(M )). It is interesting to point out that Etingof and Walton ([14]) obtained a somewhat similar result in the purely algebraic set up of finite dimensional Hopf algebra actions on commutative domains. However, their proof crucially depends on finite dimensionality of the Hopf algebra and it is not obvious whether or how it is possible to extend it for infinite dimensional Hopf algebras. On the other hand, the main theorem of the present paper will imply extension of their results for (co)-actions of Hopf-∗ algebras of compact type (but possibly infinite dimensional), or, equivalently, actions of the mutiplier Hopf algebras of the corresponding discrete quantum groups, on a large class of commutative domains arising in geometry. Combining Theorem II with the techniques developed by Bhowmick and Goswami in [8] and by Joardar-Goswami in [18], we have computed quantum isometry groups of most of the noncommutative manifolds considered in the literature, namely those coming from classical connectecd compact manifolds by suitable Rieffel-deformation or unitary twists, and obtain the following result: Theorem III (Theorem 12.7 of Section 12): The quantum isometry group of a noncommutative manifold obtained by cocycle twisting of a classical, connected, compact manifold as in [18] is a similar cocycle twisted version of the isometry group of the classical manifold. The above results are interesting because they have two-fold physical implications: firstly, it follows that for a classical mechanical system with phase-space modelled by a compact connected manifold, the generalized notion of symmetries in terms of quantum groups coincides with the conventional notion, i.e. symmetries coming from group actions. This gives some kind of consistency of the philosophy of thinking quantum group actions as symmetries. Secondly, it also allows us to describe all the (quantum) symmetries of a physical model obtained by suitable deformation of a classical model with connected compact phase space, showing that such quantum symmetries are indeed deformations of the classical (group) symmetries of the original classical model. It may be interesting to point out that in some physics literature on possible theories of quantum gravity see, e.g. [47], such deformed versions of the classical isometry 3

groups of the space-time manifold (which is not compact though) have been extensively used, and even termed as the ‘quantum isometry groups’. It is perhpas intuituively assumed there that such a deformed quantum group is the maximal quantum group acting isometrically on the deformed space-time, but we have not found a rigorous statement or proof of such assumption. Such a rigorous proof can be given if we can extend our framework of quantum isometry and the results of this paper to the set-up of locally compact quantum groups acting isometrically on locally compact noncommutative manifolds. As a biproduct, we have developed some new tools for studying quantum isometry groups of classical manifolds and more generally, actions of compact quantum groups on manifolds, which may also be useful in the more general framework of noncommutative manifolds. For example, we deduce that any smooth action on smooth compact manifold is automatically injective and admits a lift to bimodule of one-forms as an equivariant representation. Moreover, in Section 9 we have given a geometric characterization of isometric action by a CQG on compact Riemannian manifolds in terms of the Riemmanian innerproduct, proving that a CQG action is isometric if and only if it preserves the inner product. We have also extended (in Subsection 7.2) the standard trick of averaging to the framework of quantum group actions, showing how to make a smooth, possibly non-isometric CQG action α on C(M ), to be isometric w.r.t. a new ‘averaged’ Riemannian metric. We believe that all these new techniques will be extremely important for the general theory of quantum isometry groups in the long run. Let us briefly explain the strategy of proof. It hinges upon the realization that there cannot be any faithful action by a genuine CQG on a subset with nonempty interior of some Euclidean space Rn which is affine in the sense that the action leaves invariant the linear span of the coordinate functions and the constant function 1 (which satisfy so-called ‘quadratic independence in the sense of [16]). This was in fact observed in earlier works like [16] and played a crucial role in proving non-existence of genuine quantum isometries in examples studies in those papers. The key idea of the present paper for proving the main theorem , i.e. Theorem I, consists of a chain of arguments to deduce that the given CQG action must be affine on some large enough Euclidean space. To explain these steps, let us first think of a similar line of arguments in the classical set-up, i.e. a proof of the fact that any smooth action by a compact group G on a classical compact connected manifold M can be made affine in a suitable embedding of M in some Rn . This can be done as follows: (a) Using the averaging trick we reduce the problem to the case of isometric action. (b) We can assume without loss of generality that M is paralleizable and hence has an isometric embedding in some Rn with a trivial normal bundle N (M ), by passing to the paralleizable total space T (O(M )) of the orthonormal frame bundle O(M ) where the G-action on M can be lifted to the natural isometric action by taking the differential dg of the map g ∈ G. (c) We can lift the action to an isometric action on the total space T (N ) of the trivial normal bundle N (M ). 4

(d) Finally, noting that T (N ) is locally isometric to the flat Euclidean space Rn we can argue that any isometric action on it must be affine in the corresponding coordinates. The proof in the quantum case is a natural (but rather non-trivial due to noncommutativity at every stage) adaptation of the above lines of arguments in the franework of CQG action. We begin by collecting some basic facts in Sections 3,4,5 and performing the analogue of step (a) in quantum group set up in subsection 7.2. Then in Section 8 we consider the problem of lifting a smooth action of a CQG on a smooth manifold to its cotangent baundle and orthonormal frame bundle. Then in Section 9 we prove a geometric characterization of isometric action, followed by an analogue of step (c) in Section 10. Finally in section 11 observing the fact that the total space of orthonormal frame bundle is parallelizable, we complete the proof using results of previous sections. It should perhaps be mentioned that connectedness of M is used only in proving the analogue of step (d), where some standard results from the theory of harmonic function are used to deduce that the CQG action is affine.

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Notations and plan of the paper

In this paper all the Hilbert spaces are over C unless mentioned otherwise. If V is a vector space over real numbers we denote its complexification by VC . ′ For a vector space V , V stands for its algebraic dual. For a C* algebra C, M(C) will denote its multiplier algebra. ⊕ and ⊗ will denote the algebraic direct sum and algebraic tensor product respectively. We shall denote the C ∗ algebra of bounded operators on a Hilbert space H by B(H) and the C ∗ algebra of compact operators on H by B0 (H). Sp (Sp) stands for the linear span (closed linear span). Also WOT and SOT for the weak operator topology and the strong operator topology respectively. Let C be an algebra. Then σij : C ⊗ C ⊗ ... ⊗ C → C ⊗ C ⊗ ... ⊗ C is the flip map between i and j-th place | {z } {z } | n−times

n−times

and mij : C | ⊗C⊗ | ⊗C⊗ {z ... ⊗ C} → C {z ... ⊗ C} is the map obtained by multiplying n−times

(n−1)−times

i and j-th entry. In case we have two copies of an algebra we shall simply denote by σ and m for the flip and multiplication map respectively. We shall ˆ ⊗, ¯ ⊗ ¯ in (to be need several types of topological tensor products in this paper: ⊗, explained in subsequent sections). Also for a Hilbert space H and a C ∗ algebra or a locally convex ∗ algebra C we shall consider the trivial Hilbert (bi)module ¯ with the obvious right and left action of C on H⊗C ¯ coming from algebra H⊗C multiplication of C and obvious C valued inner product. When H = CN , the (bi)module is called the trivial C (bi)module of rank N . Usually, we use and for the scalar valued inner product (of a Hilbert space) and the algebravalued inner product (of a Hilbert bimodule) respectively. For a Hopf algebra H, for any C-linear map f : H → H ⊗ H, we write f (q) = q(1) ⊗ q(2) (Sweedler’s notation). For an algebra or module A and a C-linear map Γ : A → A ⊗ H, we shall also use an analogue of Sweedler’s notation and write Γ(a) = a(0) ⊗ a(1) . 5

Let us briefly sketch the plan of the paper. The paper is divided into three parts. In part I, we discuss some preliminaries. This part has three sections (section 3,4,5). In Section 3 we review the theory of tensor products of locally convex ∗-algebras as well as interior and exterior tensor products of Hilbert bimodules. In Section 4, we briefly recall the theory of compact quantum groups, their representations (both on Hilbert spaces and Hilbert bimodules) and quantum isometry groups . In Section 5, we start with discussions on the locally convex ∗-algebra C ∞ (M ) and exterior bundle as a Hilbert bimodule over it. We end Section 5 as well as Part I by quickly gathering necessary basic facts about normal bundle of an embedded Riemannian manifold. Part II is divided into 3 sections. In Section 6, we define and study smooth actions and prove in particular that a smooth action can be lifted to the space of one forms. Section 7 is devoted to discussion on inner product preserving actions. Part II ends with Section 8 where lifting of a smooth action of a CQG on a Riemannian manifold to its cotangent and orthonormal frame bundle is considered. Part III is divide into five sections. We begin Part III with a geometric characterization of an isometric action in Section 9. Section 10 deals with the problem of lifting an action on a stably parallelizable manifold to its tubular neighbourhood. Finally in Section 11, all the results of the previous sections are combined to deduce the main results of the paper. As an application we compute QISO of noncommutative manifolds obtained from classical, connected, compact manifolds by cocyle twisting. The relevant results are stated without proof in Section 12. We end Part III as well as this paper by giving some counterexamples to show that Theorem 11.9 need not hold for connected toplogical spaces which are not smooth manifolds or if we consider locally compact quantum groups which are non-compact.

Part I: Preliminaries 3 3.1

Topological tensor products Locally convex ∗ algebras

We begin by recalling from [38] the tensor product of two C ∗ algebras C1 and C2 and let us choose the minimal or spatial tensor products between two C ∗ ˆ 2 throughout algebras. The corresponding C ∗ algebra will be denoted by C1 ⊗C this paper. However we need to consider more general topological spaces and algebras. A locally convex space is a vector space equipped with a locally convex topology given by a family of seminorms. There are many ways to equip the algebraic tensor product of two locally convex spaces with a locally convex topolgy. Let E1 , E2 be two locally convex spaces with family of seminorms {||.||1,i } and {||.||2,j } respectively. Then one wants a family {||.||i,j } of seminorms for E1 ⊗ E2 such that ||e1 ⊗ e2 ||i,j = ||e1 ||1,i ||e2 ||2,j . The problem is that such a choice is far from unique and there is a maximal and a minimal choice 6

giving the projective and injective tensor product respectively. We call a locally convex space Fréchet if the family of seminorms is countable (hence the space is metrizable) and is complete with respect to the metric given by the family of seminorms. Also a Frechet locally convex space is called nuclear if its projective and injective tensor products with any other Frechet space coincide (as a locally convex space). We do not go into further details of this topic here but refer the reader to [39] for a comprehensive discussion. Furthermore if the space is a ∗ algebra then we demand that its ∗ algebraic structure is compatible with its locally convex topology i.e. the adjoint (∗) is continuous and multiplication is jointly continuous with respect to the topology. Projective and injective tensor product of two such topological ∗ algebras are again topological ∗ algebra. We shall mostly use unital ∗ algebras. Henceforth all the topological ∗-algebras will be unital unless otherwise mentioned. We now specialize to a particular class of locally convex ∗ algebras which we call ”nice“. Definition 3.1 We call a Fréchet ∗ algebra A a nice algebra if it is a ∗subalgebra of a C ∗ algebra A1 such that 1. There are finitely many densely defined closed ∗ derivations δ1 , ..., δN on A1 . 2. A ⊂ D(δi ) and A is stable under δi for all i. 3. The topology of A is given by the family of seminorms {||x||α = ||δα (x)||}, where α = (i1 , ..., ik ) : 1 ≤ ij ≤ k, k ≥ 1 is a multi index or α = φ(null index), δα = δi1 ..δik , δφ = id and ||.|| is the C ∗ norm of A1 . Given two such nice algebras A(⊂ A1 ) and B(⊂ B1 ) with finitely many derivaˆ 1 and tions {δ1 , ..., δn } and {η1 , ..., ηm } respectively, we have A ⊗ B ⊂ A1 ⊗B A ⊗ B has finitely many derivations {δ1 ⊗ id, ..., δn ⊗ id, id ⊗ η1 , ..., id ⊗ ηm }. We topologize A ⊗ B by the family of seminorms {||.||αβ } as before where this ˆ 1 . We denote time the norm on A ⊗ B is taken from the spatial norm of A1 ⊗B ˆ the completion with respect to this topology by A⊗B. It is clearly again a nice ˆ may algebra in our sense. It should be mentioned that the construction of A⊗B ∗ depend on the choice of derivations and the ambient C algebra. However if A is a nuclear algebra(for example C ∞ (M ) with its canonical locally convex topolˆ does not depend upon such choices. In this paper, we don’t usually ogy), A⊗B mention the underlying set of derivations or the ambient C ∗ algebras explicitly as they are almost always understood from the context. In fact all the locally ˆ for convex ∗ algebras considered in this paper will be of the form C ∞ (M )⊗Q ∗ some compact manifold M and some C algebra Q so that they will be ”nice” ˆ as the ambient C ∗ algebra locally convex ∗ algebras in our sense with C(M )⊗Q and the finitely many canonical derivations coming from the coordinate vector fields of the compact manifold M . Given unital nice ∗ algebras E1 , E2 , F1 , F2 and u : E1 → E2 and v : F1 → F2 be two continuous homomorphisms, the mapping u ⊗ v is continuous with respect to the locally convex topology given in the previous paragraph. We denote the continuous extension again by u⊗v. We note the following standard fact without proof which will be crucial in the analysis of smooth actions of compact quantum groups later on. 7

Proposition 3.2 If A1 , A2 , A3 are nice algebras as above and Φ : A1 × A2 → A3 is a bilinear map which is separately continuous in each of the arguments. Then Φ extends to a continuous linear map from the projective tensor product of A1 with A2 to A3 . If furthermore, A1 is nuclear, Φ extends to a continuous ˆ 2 to A3 . map from A1 ⊗A As a special case, suppose that A2 , A2 are subalgebras of a nice algebra A and also that A1 is isomorphic as a Frechet space to some nuclear space. Then ˆ 2 the multiplication map say m of A extends to a continuous map from A1 ⊗A to A.

3.2

Hilbert bimodules: interior and exterior tensor products

Let E1 and E2 be two Hilbert bimodules over two locally convex ∗ algebras(nice) C1 and C2 respectively. We denote the algebra valued inner product for the Hilbert bimodules by . When the bimodule is a Hilbert space, we denote the corresponding scalar valued inner product by . Then E1 ⊗ E2 has an ′ ′ ′ obvious C1 ⊗ C2 bimodule structure, given by (a ⊗ b)(e1 ⊗ e2 )(a ⊗ b ) = ae1 a ⊗ ′ ′ ′ be2 b for a, a ∈ C1 , b, b ∈ C2 and e1 ∈ E1 , e2 ∈ E2 . Also define C1 ⊗ C2 valued inner product by >=> ⊗ > for ¯ 2 . In fact e1 , f1 ∈ E1 and e2 , f2 ∈ E2 . We denote the completed module by E1 ⊗E ¯ 2 is an C1 ⊗C ˆ 2 bimodule. This is called the exterior tensor product of two E1 ⊗E bimodules. In particular if one of the bimodule is a Hilbert space H (bimodule over C) and the other is a C ∗ algebra Q (bimodule over itself), then the exterior ¯ tensor product gives the usual Hilbert Q module H⊗Q. When H = CN , we have a natural identification of an element T = ((Tij )) ∈ MN (Q) with the right Q linear map of CN ⊗ Q given by ei 7→ ej ⊗ Tji , where {ei }i=1,...,N is a basis for CN . We can take tensor products of maps between two Hilbert bimodules under. We shall need such tensor product of maps, which are ”isometric“ in some sense. Let Ti : Ei → Fi , i = 1, 2 be two C-linear maps and Ei , Fi are Hilbert bimodules over Ci , Di (i = 1, 2) respectively. Moreover, suppose that >= αi >, ξi , ηi ∈ Ei where αi : Ci → Di are ∗-homomrphisms. Then it is easy to show that the algebraic tensor product T := T1 ⊗alg T2 also satisfies >= (α1 ⊗ α2 )(>) and hence extends to a well defined continuous map from E1 ⊗E ¯ 2 again to be denoted by T1 ⊗ T2 . F1 ⊗F Let B, C, D be three locally convex ∗ algebras. Also let E1 be an B−C Hilbert bimodule E2 be a C −D Hilbert bimodule. Then E1 ⊗C E2 is an B −D bimodule in the usual way. We can define a D valued inner product that will make E1 ⊗C E2 a pre-Hilbert B −D bimodule. For that take ω1 , ω2 ∈ E1 and η1 , η2 ∈ E2 and define >:= η2 >> .

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Let I = {ξ ∈ E1 ⊗C E2 such that >= 0}. Then define E1 ⊗in E2 = E1 ⊗C E2 /I. We note that this semi inner product is actually an inner product, so that I = {0} (see proposition 4.5 of [22]). The topological completion of ¯ in E2 . E1 ⊗in E2 is called the interior tensor product and we shall denote it by E1 ⊗ We denote the projection map from E1 ⊗C E2 to E1 ⊗in E2 by π. We also make the convention of calling a Hilbert A − A bimodule simply Hilbert A bimodule.

4 4.1

Compact quantum groups, their representations and actions Definition and representations of compact quantum groups

A compact quantum group (CQG for short) is a unital C ∗ algebra Q with a ˆ such that each of the linear coassociative coproduct (see [24]) ∆ from Q to Q⊗Q ˆ spans of ∆(Q)(Q ⊗ 1) and that of ∆(Q)(1 ⊗ Q) is norm-dense in Q⊗Q. From this condition, one can obtain a canonical dense unital ∗-subalgebra Q0 of Q on which linear maps κ and ǫ (called the antipode and the counit respectively) are defined making the above subalgebra a Hopf ∗ algebra. In fact, this is the algebra generated by the ‘matrix coefficients’ of the (finite dimensional) irreducible non degenerate representations (to be defined shortly) of the CQG. The antipode is an anti-homomorphism and also satisfies κ(a∗ ) = (κ−1 (a))∗ for a ∈ Q0 . It is known that there is a unique state h on a CQG Q (called the Haar state) which is bi invariant in the sense that (id⊗h)◦∆(a) = (h⊗id)◦∆(a) = h(a)1 for all a. The Haar state need not be faithful in general, though it is always faithful on Q0 at least. Given the Hopf ∗-algebra Q0 , there can be several CQG’s which have this ∗-algebra as the Hopf ∗-algebra generated by the matrix elements of finite dimensional representations. We need two of such CQG’s: the reduced and the universal one. By definition, the reduced CQG Qr is the image of Q in the GNS representation of h, i.e. Qr = πr (Q), πr : Q → B(L2 (h)) is the GNS representation. There also exists a largest such CQG Qu , called the universal CQG corresponding to Q0 . It is obtained as the universal enveloping C ∗ algebra of Q0 . We also say that a CQG Q is universal if Q = Qu . We recall the universal quantum groups as in [42],[43] and references therein. For an n × n positive invertible matrix Q = (Qij ). let Au (Q) be the compact quantum group defined and studied in [40],[41] which is the universal C ∗ algebra Q generated by {uQ kj : k, j = 1 . . . n} where u := ((ukj )) satisfies ′

′

uu∗ = u∗ u = In , u Q¯ uQ−1 = Q¯ uQ−1 u . ′ ˜ is given by Here u = ((uji )) and u ¯ = ((u∗ij )). The coproduct say ∆ ˜ Q) = ∆(u ij

n X

k=1

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Q uQ ik ⊗ ukj .

(1)

It may be noted that Au (Q) is the universal object in the category of compact quantum groups which admit a unitary representation on the finite dimensional C ∗ algebra Mn (C) which preserves the functional x(∈ Mn ) → T r(QT x) (see [42]). It is also clear from the definition that Au (Q)’s are indeed universal compact quantum groups in the sense discussed in the preceding paragraph. ˆ Let H be a Hilbert space. Consider the multiplier algebra M(B0 (H)⊗Q). ˆ ⊗Q). ˆ This algebra has two natural embeddings into M(B0 (H)⊗Q The first one is obtained by extending the map x 7→ x ⊗ 1. The second one is obtained by composing this map with the flip on the last two factors. We will write ˆ w12 and w13 for the images of an element w ∈ M(B0 (H)⊗Q) by these two ˆ maps respectively. Note that if H is finite dimensional then M(B0 (H)⊗Q) is isomorphic to B(H) ⊗ Q (we don’t need any topological completion). Definition 4.1 Let (Q, ∆) be a CQG. A unitary representation of Q on a ¯ such Hilbert space H is a C-linear map U from H to the Hilbert module H⊗Q that 1.>=< ξ, η > 1Q , where ξ, η ∈ H. 2. (U ⊗ id)U = (id ⊗ ∆)U ˜ belonging to Given such a unitary representation we have a unitary element U ˜ ˆ M(B0 (H)⊗S) given by U (ξ ⊗ b) = U (ξ)b, (ξ ∈ H, b ∈ S) satisfying ˜ =U ˜ 12 U ˜ 13 . (id ⊗ ∆)U Definition 4.2 A closed subspace H1 of H is said to be invariant if U (H1 ) ⊂ ¯ H1 ⊗Q. A unitary represenation U of a CQG is said to be irreducible if there is no proper invariant subspace. It is a well known fact that every irreducible unitary representation is finite dimensional. ˆ the set of inequivalent irreducible unitary representations We denote by Q ˆ of Q. For π ∈ Q, let dπ and {tπjk : j, k = 1, ..., dπ } be the dimension and matrix coefficients of the corresponding finite dimensional representation respectively. ˆ we have a unique dπ × dπ complex matrix Fπ such that Then for each π ∈ Q, (1)Fπ is positive and invertible with T r(Fπ ) = T r(Fπ−1 ) = Mπ > 0(say). ∗ (2)h(tπij tπkl ) = M1π δik Fπ (j, l). ˆ let ρπsm be the linear functional on Q given by Corresponding to π ∈ Q, π π ρsm (x) = h(xsm x), s, m = 1, ..., dπ for x ∈ Q where xπsm = (Mπ )tπ∗ km (Fπ )ks . Pdπ π π Also let ρ = s=1 ρss . Let us discuss in some details a few facts about algebraic representation of Q on a vector space without any (apriori) topology i.e. Γ : K → K ⊗ Q0 and (Γ ⊗ id)Γ = (id ⊗ ∆)Γ, where K is some vector space. In this case the non degeneracy condition Sp Γ(K)(1 ⊗ Q0 ) = K ⊗ Q0 is equivalent to the condition Γǫ := (id ⊗ ǫ)Γ is identity on K. Then we have the following: 10

Proposition 4.3 Given two algebraic non degenerate representations Γ1 , Γ2 of Q on two vector spaces K and L respectively, we can define the tensor product and direct sum of the representations (to be denoted by Γ1 ⊗ Γ2 and Γ1 ⊕ Γ2 respectively) by (Γ1 ⊗ Γ2 )(k ⊗ l) := k(0) ⊗ l(0) ⊗ k(1) l(1) ) and (Γ1 ⊕ Γ2 )(k, l) := (k(0) , l(0) ) ⊗ k(1) l(1) . Then Γ1 ⊗ Γ2 and Γ1 ⊕ Γ2 are algebraic non degenerate representations of Q on the vector spaces K ⊗ L and K ⊕ L respectively. Proof: These are consequences of the simple observations that (Γ1 ⊗ Γ2 )ǫ = (Γ1ǫ ⊗ Γ2ǫ ) and (Γ1 ⊕ Γ2 )ǫ = (Γ1ǫ ⊕ Γ2ǫ ).✷ In this paper by an algebraic representation we shall mean a non degenerate representation unless otherwise stated.

4.2

Actions of compact quantum groups

Let C be a nice unital Fréchet ∗-algebra (in the sense discussed in Subsection 3.1) and Q be a compact quantum group. ˆ is said to be a topological action Definition 4.4 A C linear map α : C → C ⊗Q of Q on C if 1. α is a continuous ∗ algebra homomorphism. 2. (α ⊗ id)α = (id ⊗ ∆)α (co-associativity). ˆ in the corresponding Fréchet topology. 3. Sp α(C)(1 ⊗ Q) is dense in C ⊗Q Note that if the Fréchet algebra is a C ∗ algebra, then the definition of a topological action coincides with the usual C ∗ action of a compact quantum group. Definition 4.5 A topological action α is said to be faithful if the ∗-subalgebra of Q generated by the elements of the form (ω ⊗ id)α, where ω is a continuous linear functional on C, is dense in Q. Let X be a compact space. Then we can consider the C ∗ action of Q on C(X). We say Q acts topologically on a compact space X if there is a C ∗ action of Q on C(X). We have the following:([20]) Proposition 4.6 If a CQG Q acts topologically and faithfully on X, where X is any compact space, then the corresponding reduced CQG Qr (which has a faithful Haar state) must be a Kac algebra. In particular the Haar state of Qr , and hence of Q is tracial. Moreover, the antipode κ is defined and norm-bounded on Qr . Now recall from subsection 4.1 the linear functional ρπ on a compact quantum group Q. Given a topological action α of Q on a “nice” unital ∗ algebra C, we can define a projection Pπ : C → C by Pπ := (id ⊗ ρπ )α (note that (id ⊗ φ)α(C) ⊂ C for all bounded linear functionals φ on Q). If we denote Im Pπ by Cπ , then we ˆ and let C0 := ker(α)⊕ Sp{Cπ : define the spectral subspace to be Sp{Cπ : π ∈ Q} ˆ π ∈ Q}. So in particular if ker(α) = {0}, then the spectral subspace coincides 11

with C0 . Along the lines of [29] and Proposition 2.2 of [37] it can be shown that Proposition 4.7 C0 is a unital ∗ subalgebra over which α is algebraic and total. Moreover C0 is the maximal subspace over which α is algebraic and C0 is dense in C in the corresponding Fréchet topology. We end this subsection with a discussion on an analogue of an action on a von Neumann algebra given by conjugating unitary representation. Let M ⊂ B(H) be a von Neumann algebra and U be a unitary representation of a CQG Q on H. The map adU˜ is a normal, injective ∗-homomorphism on B(H). We say adU˜ leaves M invariant if (id ⊗ φ)(adU˜ (x)) ∈ M for all bounded linear functionals φ on Q. In that case we can consider the spectral projections Pπ = (id ⊗ ρπ ) ◦ adU˜ and define Mπ := Pπ (M). Then we have the spectral subalgebra Sp {Mπ , π ∈ ˆ And as ad ˜ is one-one, the spectral subalgebra is the maximal subspace of Q}. U M over which adU˜ is algebraic. We denote this subalgebra by M0 and it can be proved that M0 is SOT dense in M. We end this subsection with a discussion on the unitary implementability of an action. Definition 4.8 We call an action α of a CQG Q on a unital C ∗ algebra C to be implemented by a unitary representation U of Q in H, say, if there is a faithful ˜ (π(x) ⊗ 1)U ˜ ∗ = (π ⊗ id)(α(x)) for all representation π : C → B(H) such that U x ∈ C. It is clear that if an action is implemented by a unitary representation then it is one-to-one. In fact, as (id ⊗ πr )(U ) gives a unitary representation of Qr in H and the ‘reduced action’ αr := (id ⊗ πr ) ◦ α of Qr is also implemented by a unitary representation, it follows that even αr is one-to-one. We see below that this is actually equivalent to implementability by unitary representation: Lemma 4.9 Given an action α of Q on a unital separable C ∗ algebra C the following are equivalent: (a) There is a faithful positive functional φ on C which is invariant w.r.t. α, i.e. (φ ⊗ id)(α(x)) = φ(x)1Q for all x ∈ C. (b) The action is implemented by some unitary representation. (c) The reduced action αr of Qr is injective. Proof: If (a) holds, we consider H to be the GNS space of the faithful positive functional φ. The GNS representation π is faithful, and the linear map U defined ˆ is an isometry by the invariby U (x) := α(x) from C ⊂ H = L2 (C, φ) to H⊗Q ance of φ. Thus U extends to H and it is easy to check that it gives a unitary representation which implements α. We have already argued (b) ⇒ (c), and finally, if (c) holds, we choose any faithful state say τ on the separble C ∗ algebra C and take φ(x) = (τ ⊗h)(αr (x)), 12

which is faithful as h is faithful on Qr . It can easily be verified that φ is αinvariant on the dense subalgebra C0 mentione before, and hence on the whole of C.✷

4.3

Representation of CQG on a Hilbert bimodule over a nice topological ∗-algebra

We now generalize the notion of unitary representation on Hilbert spaces to another direction, namely on Hilbert bimodules over a nice, unital topological ∗-algebras. Let E be a Hilbert C − D bimodule over topological ∗-algebras C and D and let Q be a compact quantum group. If we consider Q as a bimodule over ¯ which is a C ⊗Q−D ˆ ˆ itself, then we can form the exterior tensor product E ⊗Q ⊗Q ˆ and αD : D → D⊗Q ˆ be topological actions bimodule. Also let αC : C → C ⊗Q ¯ of C and D on Q in the sense discussed earlier. Then using α we can give E ⊗Q ′ ′ ¯ and a C − D bimodule structure given by a.η.a = αC (a)ηαD (a ), for η ∈ E ⊗Q ′ a ∈ C, a ∈ D (but without any D valued inner product). ¯ is said to be an αD equivariant Definition 4.10 A C-linear map Γ : E → E ⊗Q unitary representation of Q on E if 1.Γ(ξd) = Γ(ξ)αD (d) and Γ(cξ) = αC (c)Γ(ξ)) for c ∈ C, d ∈ D. ′ ′ ′ 2. >= αD (>), for ξ, ξ ∈ E. 3. (Γ ⊗ id)Γ = (id ⊗ ∆)Γ (co associativity) ¯ (non degeneracy). 4. Sp Γ(E)(1 ⊗ Q) = E ⊗Q In the definition note that condition (2) allows one to define (Γ ⊗ id). Given an α equivariant representation Γ of Q on a Hilbert bimodule E, proceeding as in subsection 4.2, we can get spectral decomposition of E. We have Eπ := Im Pπ (= ˆ ⊕ ker(Γ). In case Γ is one-one which (id ⊗ ρπ )Γ). Define E0 := Sp{Eπ : π ∈ Q} is equivalent to α being one-one, E0 coincides with the spectral subspace. Again proceeding along the lines of [29] and [37], we can prove the following Proposition 4.11 1. (id ⊗ φ)Γ(E) ⊂ E, for all bounded linear functional φ ∈ Q∗ . 2. Pπ2 = Pπ . 3. Γ is algebraic over E0 i.e. Γ(E0 ) ⊂ E0 ⊗ Q0 and (id ⊗ ǫ)Γ = id on E0 . Also there is a maximal subspace E˜ over which Γ is algebraic. 4. E0 is dense in E and hence E˜ is dense in E. Lemma 4.12 Let E1 be a Hilbert B − C bimodule and E2 be a Hilbert C − D bimodule. αB ,αC ,αD be topological actions on a compact quantum group Q of ¯ and Γ2 : E2 → E2 ⊗Q ¯ topological ∗-algebras B, C, D respectively. Γ1 : E1 → E1 ⊗Q be αC and αD equivariant unitary representations as discussed earlier. Then ′

′

′

′

Γ2 (η ) >>= αD η >> . 13

Proof: ′

′

Γ2 (η ) >> ′

′

=

)Γ2 (η ) >>

=

η ) >>

=

αD η >>

′

′

′

′

✷ ¯ Theorem 4.13 Given an αD equivariant unitary representation Γ : E → E ⊗Q of a CQG Q on a Hilbert C − D bimodule E. Let E0 , C0 and D0 be as defined earlier. Then E0 is a Hilbert C0 − D0 bimodule. Proof: Note that E0 is dense in E and E0 is the maximal C linear subspace over which Γ is algebraic. Similarly we have dense subalgebras C0 of C and D0 of D over which αC and αD are algebraic. C0 and D0 are also maximal subspaces over which αC and αD are algebraic. Let e ∈ E0 and a ∈ D0 . Then Γ(ea) = Γ(e)αD (a) = e(0) a(0) ⊗ e(1) a(1) , where Γ(e) = e(0) ⊗ e(1) and αD (a) = a(0) ⊗ a(1) . e(0) ∈ E0 and a(0) ∈ D0 implies Γ(E0 D0 ) ⊂ E0 D0 ⊗ Q0 , that is Γ is algebraic on C-linear subspace E0 D0 and hence by maximality E0 D0 ⊂ E0 . Similarly we can show that C0 E0 ⊂ E0 . ′ ′ ′ Finally let F = Sp {> |e, e ∈ E0 } and for e, e ∈ E0 we have ′

αD (>) ′

= > (by αD equivariance) ′

′

= > ⊗e∗(1) e(1) ⊂ F ⊗ Q0 Again by maximality we have F ⊂ D0 . Hence indeed E0 is a Hilbert C0 − D0 bimodule.✷ We can form direct sum and tensor product of equivariant representations extending the algebraic constructions. We have Lemma 4.14 Let Γ1 , ..., Γk be α equivariant representations on the Hilbert A bimodules E1 , ..., Ek respectively. Then Γ1 ⊕ ... ⊕ Γk is again an α equivariant representation on the direct sum bimodule E1 ⊕ ... ⊕ Ek . Proof: Follows from the definition and 4.3.✷

14

Lemma 4.15 Let E1 , E2 , B, C, D, αB , αC , αD , Γ1 , Γ2 , Q be as in Lemma 4.12. By Theorem 4.13, we have dense subspaces Ei0 of Ei for i = 1, 2 and dense ∗subalgebras B0 , C0 , D0 such that E10 is a Hilbert B0 − C0 bimodule and E20 is a Hilbert C0 − D0 bimodule. Then we have an αD equivariant representation Γ of ¯ in E2 . Q on the Hilbert B − D bimodule E1 ⊗ Proof: Recall from subsection on algebraic representations of CQG on vector spaces, the representation Γ1 ⊗Γ2 of Q on E10 ⊗E20 . The non degeneracy of Γ1 and Γ2 implies the non degeneracy of Γ1 ⊗ Γ2 . Also applying Lemma 4.12, it is easy to see that (Γ1 ⊗ Γ2 ) ◦ π = (π ⊗ idQ )(Γ1 ⊗ Γ2 ) on E10 ⊗ E20 , where π : E10 ⊗ E20 → E10 ⊗in E20 is the projection map as in subsection 2.2. Hence Γ1 ⊗ Γ2 descends to an algebraic representation of Q on E10 ⊗in E20 . Lemma 4.12 also implies the αD ¯ in E2 and density of equivariance of Γ1 ⊗ Γ2 . So by density of E10 ⊗in E20 in E1 ⊗ ¯ in E2 ⊗Q, ¯ E10 ⊗in E20 ⊗ Q0 in E1 ⊗ we get the desired Γ. ✷ In particular when E is the trivial C-bimodule of rank N , we have the following: N Lemma 4.16 Given PN an α equivariant representation Γ of Q on C ⊗ C such ˆ for all i, j = 1, ..., N , where {ei ; i = that Γ(ei ⊗ 1C ) = j=1 ej ⊗ bji , bij ∈ C ⊗Q 1, ..., N } is an orthonormal basis of CN , then U = ((bij ))i,j=1,....,N is a unitary ˆ element of MN (C ⊗Q).

Proof: Since Γ is α-equivariant, we have >= α(>) ⇒

N X

>= α(δij 1C )

k,l=1

⇒

N X

b∗ki bkj = δij 1C ⊗Q ˆ

k=1

ˆ Hence U ∗ U = 1MN (C ⊗Q) , i.e. U is a partial isometry in MN (C ⊗Q). Viewing ˆ N ˆ linear map on the trivial module (C ⊗Q) ˆ U as a right C ⊗Q , it is enough ˆ N . This follows from density of to show that range of U is dense in (C ⊗Q) N N ˆ and observing that Sp Γ(C ⊗ C)(1 ⊗ Q) = C ⊗ C ⊗Q

=

Γ(ei ⊗ a)(1 ⊗ q) X ej ⊗ bji α(a)(1 ⊗ q) j

= ⊂

U (ei ⊗ 1C ⊗Q ˆ )α(a)(1 ⊗ q) R(U )

for a ∈ C, q ∈ Q. ✷ Now we turn to the concept of symmetric algebra of a module over a unital 15

commutative algebra. Let W be a (bi)module over a unital commutative algebra say M and let c.w = w.c for w ∈ W and c ∈ M. Then the symmetric algebra corresponding to W is by definition the algebraic of symmetric tensor Ldirect sum sym sym , where W(0) := M products (over M) of copies of E, i.e. S(W) := n≥0 W(n) sym and for n ≥ 1, W(n) is the symmetric tensor product of n copies (which is possible to define as M is commutative). The M linear span of elements of the form e ⊗...⊗ e for e ∈ W coincides with Wnsym . This has a natural algebra | {z } n−times

structure coming from tensor multiplication. Moreover, it has the following universal properties: Proposition 4.17 Given any unital commutative algebra G with an algebra inlusion i (unital), which naturally gives G an M ≡ i(M)-module structure, and an M-module map θ : W → G, there is a unique lift θˆ : S(W) → G satisfying ˆ W sym = θ, which is an algebra homomorphism. Moreover, ˆ W sym = i and θ| θ| (1) (0) if G is a ∗-algebra and if there is an antilinear involution say w 7→ w on W, with θ(w) = θ(w)∗ , then we can define a ∗-algebra structure on S(W) satisfying (w1 ⊗ w2 ⊗ . . . wn )∗ = w n ⊗ w n−1 ⊗ . . . ⊗ w 1 , and θˆ is ∗-homomorphism w.r.t. this structure.

Now let E be a Hilbert C bimodule, where C is a nice commutative locally convex ∗ algebra such that c.ξ = ξ.c, for ξ ∈ E and c ∈ C. Also let A2 be another nice commutative locally convex ∗ algebra such that i : C → A2 is a ∗ homomorphism and θ : E → A2 be a continuous module map. Then we have the following: ˆ ˆ Lemma 4.18 (i) If E1 is a dense submodule of E, θ(S(E 1 )) is dense in θ(S(E)). (ii) Let there be an αC -equivariant representation Γ of a CQG Q on E and suppose that F is a submodule of E such that Γ restricts to an αC equivariant algebraic representation of Q (as discussed earlier) on F such that {>: e, f ∈ F } is a commutative subalgebra of F ⊗ Q0 , then we have an sym sym αC equivariant coassociative map Γs(n) : F(n) :→ F(n) ⊗ Q0 , hence also a map s ˆ : S(F ) → S(F ) ⊗ Q0 such that Γ = Γ on F . Γ (1)

Proof: Pl The proof of (i) is straightforward. For (ii) let e ∈ F and Γ(e) = i=1 ei ⊗ qi . Pl Then Γ(n) (e ⊗ e ⊗ ... ⊗ e) = i1 ,...,in=1 ei1 ⊗ ei2 ⊗ ... ⊗ ein ⊗ qi1 qi2 ...qin . Also sym let P : F ⊗ F ⊗ ... ⊗ F → F(n) be the projection map onto the symmetric submodule. Let f1 ⊗ f2 ⊗ ... ⊗ fn ∈ F ⊗ F ⊗ ... ⊗ F . Then > =

l X X 1 > ... > qi1 qi2 ...qin n! i ,...,i =1 1

=

1 X n!

n

σ∈Sn

l X

> ... > qi1 qi2 ...qin

σ∈Sn i1 ,...,in =1

16

But by the hypothesis, for all σ ∈ Sn > ... >=> ... > l X

⇒

> ... > qi1 qi2 ...qin =

i1 ,...,in =1 l X

> ... > qi1 ...qin

i1 ,...,in =1

Hence > l X 1 > ... > qi1 ...qin n! i ,...,i =1

=

n!

=

>

1

n

Now f1 , ..., fn ∈ F being arbritary we can conclude that (P ⊗idQ0 )(Γ(n) (e⊗...⊗ sym sym e)) = Γ(n) (e ⊗ ... ⊗ e), that is, Γ(n) maps F(n) into F(n) ⊗ Q0 , hence induces s the required map Γ(n) . ✷

4.4

Quantum isometry groups

Let us briefly sketch the formulation of quantum isometry groups of spectral triples given by Goswami and Bhowmick in [11], [15]. Definition 4.19 Let (A∞ , H, D) be a spectral triple of compact type (a la Connes). Consider the category Q(D) ≡ Q(A∞ , H, D) whose objects are triples (Q, ∆, U ) where (Q, ∆) is a CQG having a unitary representation U on the Hilbert space H ˜ commutes with (D ⊗ 1Q ). Morphism between two such objects (Q, ∆, U ) and U ′ ′ ′ ′ ′ and (Q , ∆ , U ) is a CQG morphism ψ : Q → Q such that U = (id ⊗ ψ)U . Moreover for a positive (possibly unbounded) operator R on H which commutes ′ ′ with D, consider the subcategory QR (D) ≡ QR (A∞ , H, D) whose objects are triples (Q, ∆, U ) as before with the additional requirement that adU˜ preserves the functional τR (x) = T r(Rx)(x ∈ ED ) in the sense that (τR ⊗ id)adU˜ (x) = τR (x).1Q , where ED be the weakly dense ∗-subalgebra of B(H) spanned by the rank one operators of the form |ξ >< η| where ξ and η are eigen vectors of D. ′

Definition 4.20 The universal object in the category QR (D) exists and is de(A∞ , H, D). However if U is the corresponding unitary repnoted by QISO+^ R

resentation, adU˜ may not be faithful. The largest Woronowicz subalgebra of +^ QISOR (A∞ , H, D) on which adU˜ is faithful is called quantum group of orientation preserving Riemannian isometry of the R-twisted spectral triple and denoted 17

+ + by QISOR (A∞ , H, D) or simply by QISOR (D). Q (D) may not have a universal object in general, but if it exists we shall denote it by QISO+^ (A∞ , H, D) + ∞ and the corresponding largest Woronowicz subalgebra by QISO (A , H, D) or simply QISO+ (D). ′

We note that in case of quantum group of orientation preserving Riemannian ′′ + isometry, QISOR (D) really depends on the von Neumann algebra (A∞ ) , not on the algebra A∞ itself. More precisely, we have the following proposition, + ^ which follows from the definition of QISO (D) and needs no proof: R

∞

Proposition 4.21 Let (A , H, D, R) be as in [11]. If we have a SOT dense ′′ subalgebra A0 of (A∞ ) ⊂ B(H) such that [D, a] ∈ B(H) for all a ∈ A0 , then (A0 , H, D) is again a spectral triple and + +^ ^ QISOR (A0 , H, D) ∼ (A∞ , H, D). = QISOR + + and hence QISOR (A0 , H, D) ∼ (A∞ , H, D). = QISOR

In [15], for a spectral triple satisfying some mild regularity conditions, a noncommutative analogue of the Hodge Laplacian was constructed which is denoted by L = LD . we denote the category whose objects are triples (S, ∆, α) where (S, ∆) is a CQG and α is an action of the CQG on the manifold com′ muting with the Laplacian (as in the sense of definition 2.11 of [15]) by QL and morphism between two such objects is a CQG morphism intertwining the actions (see [15]). It is shown in [15] that under certain regularity conditions (see [15]) on the spectral triple universal object exists in this category and we denote the universal object by QISOL and call it the quantum isometry group of the spectral triple.

5 5.1

Locally convex ∗ algebras and Hilbert bimodules coming from classical geometry C ∞ (M) as a nice algebra

All the notations are as in the subsection 3.1. Let M be a smooth n- dimensional compact Riemannian manifold possibly with boundary. We denote the algebra of real (complex) valued smooth functions on M by C ∞ (M )R (C ∞ (M )). Clearly C ∞ (M ) is the complexification of C ∞ (M )R . We also equip it with a locally convex topology : we say a sequence fn ∈ C ∞ (M ) converges to an f ∈ C ∞ (M ) if for every compact set K within a single coordinate neighborhood (M being compact, has finitely many such neighborhoods) and a multi index α, ∂ α fn → ∂ α f uniformly over K. Equivalently, let U1 , U2 , ..., Ul be a finite cover of M . Then it is a locally convex topology described by a countable family of seminorms given by: pK,α = sup |∂ α f (x)|, i x∈K

18

where K is a compact set within Ui , α is any multi index, i = 1, 2, ....l. C ∞ (M ) is complete with respect to this topology (example 1.46 of [34] with obvious modifications) and hence this makes C ∞ (M ) a locally convex Frechet ∗ algebra with obviuos ∗ structure. Note that, by choosing a finite C ∞ partition of unity on the compact manifold M , we can obtain finite set {δ1 , ..., δN } for some N ≥ n of globally defined vector fields on M which is complete in the sense that {δ1 (m), ..., δN (m)} spans Tm (M ) for all m (need not be a basis). It follows that the locally convex topology on C ∞ (M ) is given by the derivations {δ1 , ..., δN } making it a nice algebra in the sense of subsection 3.1 and this topology is the same as the Fréchet topology of C ∞ (M ) described earlier. With this topology C ∞ (M ) is nuclear (see example 6.2 of [28] ). Let E be any locally convex space. Then we can define the space of E valued smooth functions on a compact manifold M . Take a centered coordinate chart (U, ψ) around a point x ∈ M . Then an E valued function f on M is said to be smooth at x if f ◦ ψ −1 is smooth E valued function at 0 ∈ Rn in the sense of [39](definition 40.1). We denote the space of E valued smooth functions on M by C ∞ (M, E). We can give a locally convex topology on C ∞ (M, E) by the family of seminorms given by pK,α (f ) := supx∈K ||∂ α f (x)||, where i, K, α are as i before. Then we have the following Proposition 5.1 1. If E is complete, then so is C ∞ (M, E). ˆ ∼ 2. Suppose E is a complete locally convex space. Then we have C ∞ (M )⊗E = C ∞ (M, E). 3. Let M and N be two smooth compact manifolds with boundary. Then ˆ ∞ (N ) ∼ C ∞ (M )⊗C = C ∞ (M × N ) and contains C ∞ (M ) ⊗ = C ∞ (M, C ∞ (N )) ∼ ∞ C (N ) as Frechet dense subalgebra. ˆ ∼ 4. Let Q be a C ∗ algebra. Then C ∞ (M )⊗Q = C ∞ (M, Q) as C ∗ algebra. For the proof of the first statement see 44.1 of [39]. The second statement also follows from 44.1 of [39] and the fact that C ∞ (M ) is nuclear. In particular the isomorphism does not depend on the choice of derivations. The third and fourth statements follow from the second statement (replacing E suitably). ′ Now let Q and Q be two C ∗ algebras. The locally convex ∗ algebra ˆ ∼ C ∞ (M )⊗Q = C ∞ (M, Q) ′ is nice but not nuclear in general. Thus the tenˆ sor product C ∞ (M, Q)⊗Q may depend on our choice of derivations. However we will make the following convention: Choose any complete set of smooth vector fields {δ1 , ..., δN } on M as before ˆ and take {δ1 ⊗ idQ , ...δN ⊗ idQ } as the set of derivations on C ∞ (M )⊗Q. Similarly consider {δ ⊗ idQ⊗Q ˆ ′ , ..., δN ⊗ idQ⊗Q ˆ ′ } as the set of derivations for taking ′ ˆ ⊗Q ˆ ). Then we have the tensor product C ∞ (M )⊗(Q ′ ′ ′ ∼ ˆ ˆ ⊗Q ˆ )∼ ˆ ) Lemma 5.2 C ∞ (M, Q)⊗Q = C ∞ (M )⊗(Q = C ∞ (M, Q⊗Q

Proof: The first isomorphism follows from our choices of derivations and the last isomorphism follows from the nuclearity of C ∞ (M ) and hence in particular none 19

of the isomorphism depends on the choice of derivations on M .✷

5.2 5.2.1

Hilbert bimodule structure on exterior bundles Riemannian structure and C ∞ (M )-valued inner product on one-forms

Let M be a compact smooth manifold. Also let Λk (C ∞ (M )) be the space of smooth k forms on the manifold M . We equip Λ1 (C ∞ (M )) with the natural locally convex topology induced by the locally convex topology of C ∞ (M ) given by a family of seminorms {p(U,(x1 ,...,xn ),K,β) }, where (U, (x1 , . . . , xn )) is a local cordinate chart, β = (β1 , β2 , . . . , βr ) is a multi-index with αi ∈ {1, 2, . . . , n} as before, K is a compact subset, and p(U,(x (ω) := supx∈K,1≤i≤n |∂β fi (x)|, n ),K,β) P1 ,...,x n where fi ∈ C ∞ (M ) such that ω|U = i=1 fi dxi |U . It is clear from the definition that the differential map d : C ∞ (M ) → Ω1 (C ∞ (M )) is Fréchet continuous. Lemma 5.3 Let A be a Fréchet dense subalgebra of C ∞ (M ). Then Λ1 (A) := Sp {f dg : f, g ∈ A} is dense in Λ1 (C ∞ (M )). Proof: It is enough to approximate f dg where f, g ∈ C ∞ (M ) by elements of Ω1 (A). By Fréchet density of A in C ∞ (M ) we can choose sequences fm , gm ∈ A such that fm → f and gm → g in the Fréchet topology, hence by the continuity of d and the C ∞ (M ) module multiplication of Λ1 (C ∞ (M )), we see that fm dgm → f dg in Λ1 (C ∞ (M )).✷ Let Ωk (C ∞ (M ))u be the space of universal k-forms on the manifold M and δ be the derivation for the universal algebra of forms for C ∞ (M ) i.e δ : Ωk (C ∞ (M ))u → Ωk+1 (C ∞ (M ))u (see [23] for further details). By the universal property ∃ a surjective bimodule morphism π ≡ π(1) : Ω1 (C ∞ (M ))u → Λ1 (C ∞ (M )), such that π(δg) = dg. Ω1 (C ∞ (M ))u has a C ∞ (M ) bimodule structure: n n X X f gi δhi gi δhi ) = f( i=1

i=1

n X

(

i=1

gi δhi )f =

n X

(gi δ(hi f ) − gi hi δf )

i=1

As M is compact, there is a Riemannian structure. Using the Riemannian structure Ω1 (C ∞ (M )) with a C ∞ (M ) valued inner product Pn on M we Pncan ′equip ′ >∈ C ∞ (M ) by the following prescription: for x ∈ M choose a coordinate nighborhood (U, x1 , x2 , ...., xn ) around x such

20

that dx1 , dx2 , ..., dxn is an orthonormal basis for Tx∗ M . Note that the topology does not depend upon any particular choice of the Riemannian metric. Then
(x) = (

i=1

X

i,j,k,l

′

¯ i ∂gj ′ ∂g f¯i fj ( ))(x) ∂xk ∂xl

We see that a sequence ωn → ω in Λ1 (C ∞ (M )) if >→ 0 in Fréchet topology of C ∞ (M ). With this Λ1 (C ∞ (M )) becomes a Hilbert module. We need to consider smoothly varying semi-inner product on tangent spaces, which is a generalization of Riemannian structure. Definition 5.4 Let us call a smooth assignment M ∋ p 7→< ·, · >p of posibbly degenerate nonnegative bilinear forms on Tp∗ (M ) (in other words semi-inner product on each fibre of T ∗ M ) a (possibly degenerate) semi-Riemannian structure. Clearly, any such assignment gives a nonnegative C ∞ (M )-linear C ∞ (M )valued form > on Ω1 (C ∞ (M )) given by > (p) :=< w(p), w′ (p) >p . For a ∗-subalgebra A of C ∞ (M ), we say that the semi-Riemannian structure is faithful or non-degenerate on Ω1 (A) (or simply faithful on A) if >= 0, w ∈ Ω1 (A) implies w = 0. We also give another algebraic description of (semi)-Riemannian structure needed later on. For this we state and prove a well-known fact. Lemma 5.5 Let δ : C ∞ (M ) → C(M ) be a derivation such that δ(f ) is actually is smooth for f in a Frechet-dense subspace say V of C ∞ (M ). Then δ(f ) = X(f ) for some smooth vector field in M . Proof : Clearly, for any p ∈ M , the map f 7→ δ(f )(p) must be of the form τ (f ) for some τ ∈ Tp M . Thus, in a coordinate neighbourhood UParound p with x1 , . . . , xm ∂ being the local coordinates, we can write δ(f )(x) = i ai (x) ∂x f (x) for x ∈ U , i where each ai is continuous. Now, for f ∈ V we have ai to be smooth on U . Now, by the Frechet density of V we can choose f1 , . . . , fm in V with the prop∂ fi (p))) is nonsingular, and hence it is also nonsingular in erty that A(p) := (( ∂x j −1 some neighbourhood P say W of p. Clearly, A (·) will also have smooth entries and we have ai = j Bij δ(fj ) on W , which proves the smoothness of ai ’s on W. ✷ Proposition 5.6 (A) Let A ⊆ C ∞ (M ) be a Frechet-dense unital ∗-algebra as before. Assume also that T : A → C ∞ (M ) is a C-linear map satisfying the following: (i) [[T, Mf ], Mg ]] = Mh for some h := kT (f, g) ∈ C ∞ (M ), for all f, g ∈ A, where Mξ denotes the operator of multiplication by ξ ∈ C ∞ (M ), i.e. Mξ η = ξη. Moreover, (f, g) 7→ kT (f, g) extends to a continuous bilinear map from C ∞ (M ) × C ∞ (M ) (Frechet topology) to C(M ) (with the norm topology).

21

(ii) kT is a nonnegative definite kernel in the sense that for all f1 , . . . , fn ∈ ∞ A, ((kT (f i , fj ))) is a nonnegative element Pn of C (M ) ⊗ Mn . (iii) If Pfor g1 , . . . , gn , f1 , . . . , fn ∈ A, i,j=1 g i gj kT (f i , fj ) = 0 then we must have i gi dfi = 0. Then there is a unique (possibly degenerate) semi-Riemannian structure on M , with the assosiated C ∞ (M )-valued inner product > such that kT (f , g) => for all f, g ∈ A. This semi-Riemannian structure is faithful on A. (B) If we are also given the following condition (iv) then the above semiRiemannian structure is actually Riemannian. Pn (iv) For g1 , . . . , gn , P f1 , . . . , fn ∈ A, p ∈ M , i,j=1 gi (p))gj (p)kT (f i , fj )(p) = 0 then we must have i gi (p)dfi (p) = 0 in Tp∗ M . (C) Moreover, for any Riemannian structure on M , the associated Laplacian T = −d∗ d satisfies the above properties (i),(ii),(iii), (iv). Proof: We first observe that for fixed f , g 7→ [[T, f ], g] gives a derivation from a Frechetdense subalgebra to C ∞ (M ), and by the given continuity it extends to a derivation from C ∞ (M ) to C(M ), hence must be of the form Xf (g) for some vector field Xf by Lemma 5.5. That is, in local coordinates, say (x1 , ..., xm ) (if m Pm ∂ is the dimension of M ), we can write Xf = i=1 ηi (f ) ∂x , for some smooth i functions ηi (f ), and it is easy to show that each f 7→ ηi (f ) is also a derivation on A, which extends to C ∞ again by continuity. Again by Lemma 5.5, we get C ∞ -functions say gij defined in the neighbourhood of the local chart such that P ∂f ∂g . From the condition (ii) we see that ((gij (p))) is kT (f, g)(p) = ij gij (p) ∂x i ∂xj nonnegative definite for every p, so that it defines a semi-Riemannian structure and it follows from the condition (iii) that its is faithful on A. To prove (B), note that (iv) implies the following: if > (p) = 0, ω ∈ Ω1 (A) then ω(p) = 0. Recall that by the density of Ω1 (A) in Ω1 (C ∞ (M )), the space {ω(p), ω ∈ Ω1 (A) is also dense in the finite-dimensional vector space Tp∗ M , hence must coincide with it. Thus, for ξ ∈ Tp∗ M , we have some ω ∈ Ω1 (A) such that ξ = ω(p), so that < ξ, ξ >p => (p) = 0 implies ξ = ω(p) = 0. This proves the nonsingularity of ((gij (p))) at every p, completing the proof of (B). The verification of conditions (i)-(iv) for the Laplacian −d∗ d can be done using local coordinates, hence omitted.✷

5.2.2

Hilbert bimodule of higher forms

Let us now recall from [23] (pages 95-108) an algebraic construction of the C ∞ (M ) bimodule of k-forms Λk (C ∞ (M )) on a manifold M from the so-called universal forms. Ω2 (C ∞ (M ))u = Ω1 (C ∞ (M ))u ⊗C ∞ (M) Ω1 (C ∞ (M ))u and Ωk (C ∞ (M ))u = Ωk−1 (C ∞ (M ))u ⊗C ∞ (M) Ω1 (C ∞ (M ))u . Also Ω1 (C ∞ (M )) ≡ Λ1 (C ∞ (M )). For k ≥ 2, Ωk (C ∞ (M )) = Ωk−1 (C ∞ (M ))⊗in 22

Ω1 (C ∞ (M )).

˙ ∞ (M )) = ⊕k≥0 Ωk (C ∞ (M )). Ω(C

By the universality of Ω2 (C ∞ (M ))u , we have a surjective bimodule morphism π(2) : Ω2 (C ∞ (M ))u → Ω2 (C ∞ (M )). Let J2 be a submodule of Ω2 (C ∞ (M )) given by J2 = {π(2) (δω)|π(ω) = 0 f or ω ∈ Ω1 (C ∞ (M ))u } . In fact it is closed. Denote k

Ω2 (C ∞ (M)) J2

by Λ2 (C ∞ (M )). Sim-

∞

ilarly Λk (C ∞ (M )) = Ω (CJk(M)) where Jk = {π(k) (δω)|π(k−1) (ω) = 0 f or ω ∈ Ωk−1 (C ∞ (M ))u }. If ω and η belong to Ω1 (C ∞ (M )), sometimes we denote the image of ω ⊗ η in Ω2 (C ∞ (M )) by ωη and in Λ2 (C ∞ (M )) by ω ∧ η. Similar notations will be used for products in Ωk (C ∞ (M )) and Λk (C ∞ (M )). With this, the familiar de Rham differential is given by d : Λk (C ∞ (M )) → Λk+1 (C ∞ (M )) [π(k) (ω)] → [π(k+1) (δω)] ([ξ] := ξ + Jk f or ξ ∈ Ωk (C ∞ (M ))) Note that for all k, Ωk (C ∞ (M )) is the module of smooth sections of a Hermitian, smooth, locally trivial vector bundle Ek = Λ1 (M ) ⊗...⊗ Λ1 (M ) on | {z } k−times

M , whose fibre at x can be identified with the finite dimensional Hilbert space k Ekx := (Tx∗ M )⊗ , the inner product coming from the Riemannian structure. By construction, the closed submodule Jk is nothing but the module of smooth seck k ∼ tions of a sub bundle (say Vk ) of Ek , so that E Vk = Λ (M ). At the fibres of x ∈ M , we have the orthogonal decomposition of Hilbert spaces Ekx = Vkx ⊕ (Vkx )⊥ identifying the fibre of Λk (M ) at x with (Vkx )⊥ . So we have the following orthogonal decomposition of the Hilbert bimodule Ωk (C ∞ (M )):

Lemma 5.7 Ωk (C ∞ (M )) = Λk (C ∞ (M )) ⊕ Jk . In other words, Λk (C ∞ (M )) is an orthocomplemented closed submodule of Ωk (C ∞ (M )). We can also derive the above orthogonal decomposition in a purely algebraic way. For example for k = 2, let π(2) (δf ⊗ δg) ∈ Ω2 (C ∞ (M )). Then π(2) (δ(δ(g)f )) = −π(2) (δg ⊗ δf ). Hence π(2) (δ(f δg − δgf )) = π(2) (δf ⊗ δg + δg ⊗ δf ). But π(f δg − δgf ) = 0 in Ω1 (C ∞ (M )). So 12 π(2) (δf ⊗ δg + δg ⊗ δf ) ∈ J2 . Similarly 21 π(2) (δf ⊗ δg − δg ⊗ δf ) ∈ Λ2 (C ∞ (M )). Hence π(2) (ηδf ⊗ δg) = 21 π(2) (η(δf ⊗ δg + δg ⊗ δf )) + 12 π(2) (η(δf ⊗ δg − δg ⊗ δf )). Also by definition >= 0 For k ≥ 2, observe that the permutation group Sk naturally acts on Ωk (M ), where the action is induced by the obvious Sk -action on the finite dimensional k Hilbert space (Tx∗ M )⊗ , which permutes the copies of Tx∗ M at x ∈ M . Then k we have the orthogonal decomposition of the Hilbert space (Tx∗ M )⊗ into the

23

spectral subspaces with respect to the action of Sk . Explicitly k

k

(Tx∗ M )⊗ = ⊕χ∈Sˆk Pχx ((Tx∗ M )⊗ ), where Pχx is the spectral projection with respect to the character χ. Observe k that when χ = sgn, i.e. χ(σ) = sgn(σ), then Pχx ((Tx∗ M )⊗ ) = Λk (Tx∗ M ) and k Vkx = ⊕χ6=sgn Pχx ((Tx∗ M )⊗ ). Clearly this fibre-wise decomposition induces a similar decomposition at the Hilbert bimodule level, i.e. Ωk (C ∞ (M )) =

⊕χ6=sgn Pχ (Ωk (C ∞ (M ))) ⊕ Psgn (Ωk (C ∞ (M ))) Jk (C ∞ (M )) ⊕ Λk (C ∞ (M )),

=

where Pχ now denotes the spectral projection with respect to χ for the Sk -action on Ωk (C ∞ (M )) coming from Pχx fibre wise. Also observe that the above arguements go through if we replace C ∞ (M ) by any subalgebra A. In fact, if we are given any semi-Riemannian (possibly degenerate) structure on M which gives a nonnegative definite bilinear C ∞ valued form which is faithful (i.e. strictly positive definite) on Ω1 (A) then the action of the permutation group Sk on the k-fold tensor product Ωk (A) is inner product preserving, hence different spectral subspaces for Sk -action are mutnually orthogonal w.r.t. the C ∞ (M )-valued inner product. Thus, we have the following P Corollary 5.8 Ωk (A) = Λk (A) ⊕ JkA , where Ω1 (A) = { fi dgi , fi , gi ∈ A}, Ωk (A) = Ωk−1 (A) ⊗A Ω1 (A),

JkA = {π(k) (δω)|π(k−1) (ω) = 0 f or ω ∈ Ωk−1 (A)u } and Λk (A) =

Ωk (A) , JkA

Moreover if A is Fréchet dense in C ∞ (M ), Ωk (A), Λk (A) and JkA are dense in the Hilbert modules Ωk (C ∞ (M )), Λk (C ∞ (M )) and Jk respectively. ¯ has a natural C ∞ (M )⊗Q ˆ Now if Q is a C ∗ algebra then Λk (C ∞ (M ))⊗Q bimodule structure. The left action is given by X X X ′ ′ ( fi ⊗ qi )( [π(k) (ωj )] ⊗ qj ) = ( [π(k) (fi ωj )] ⊗ qi qj ) i

j

i,j

The right action is similarly given. The inner product is given by X X ′ X ′ ′ ′ > ⊗qi∗ qj . ωj ⊗ qj >>= ωi ⊗ qi , → 0 in C ∞ (M )⊗Q

24

5.2.3

Hodge ⋆ map

Now consider the case when M is orientable and a globally non-vanishing n-form (n being the dimension on M ) has been chosen. We introduce the Hodge star operator, which is a pointwise isometry ∗ = ∗x : Λk Tx∗ M → Λn−k Tx∗ M . Choose a positively oriented orthonormal basis {θ1 , θ2 , ..., θn } of Tx∗ M . Sincs ∗ is a linear transformation it is enough to define ∗ on a basis element θi1 ∧ θi2 ∧ ... ∧ θik (i1 < i2 < ... < ik ) of Λk Tx∗ M . Note that dvol(x)

= =

p det(< θi , θj >)θ1 ∧ θ2 ∧ ... ∧ θn θ1 ∧ θ2 ∧ ... ∧ θn

Definition 5.9 ∗(θi1 ∧ θi2 ∧ ... ∧ θik ) = θj1 ∧ θj2 ∧ ... ∧ θjn−k where θi1 ∧ θi2 ∧ ... ∧ θik ∧ θj1 .. ∧ θjn−k = dvol(x). Since we are using C as the scalar field, we would like to define ω ¯ for a k form ω. In the set-up introduced just before the definition we have some scalars P ¯ to be ci1 ,...,ik P such that ω(x) = ci1 ,...,ik θi1 ∧ θi2 ∧ ... ∧ θik . Then define ω ω ∧ ∗η) ω ¯ (x) = c¯i1 ,...,ik θi1 ∧ θi2 ∧ ... ∧ θik . Then the equation >= ∗(¯ defines an inner product on the Hilbert module Λk (C ∞ (M )) for all k = 1, ..., n which is the same as the C ∞ (M ) valued inner product defined earlier. Then the Hodge star operator is a unitary between two Hilbert modules Λk (C ∞ (M )) and Λn−k (C ∞ (M )) i.e. >=>. For further details about the Hodge star operator we refer the reader to [33]. Hence we have (∗ ⊗ id) : Λk (C ∞ (M )) ⊗ Q → Λn−k (C ∞ (M )) ⊗ Q. Since Hodge ∗ operator is an isometry, (∗ ⊗ id) is continuous with respect to ˙ ∞ (M ))⊗Q. ˆ the Hilbert module structure of Λ(C So we have ¯ → Λn−k (C ∞ (M ))⊗Q. ¯ (∗ ⊗ id) : Λk (C ∞ (M ))⊗Q ¯ → Λn−k (C ∞ (M ))⊗Q ¯ We derive a characterization for (∗⊗id) : Λk (C ∞ (M ))⊗Q for all k = 1, ..., n. ¯ and X ∈ Λk (C ∞ (M ))⊗Q. ¯ Lemma 5.10 Let ξ ∈ Λn−k (C ∞ (M ))⊗Q Then the following are equivalent: ¯ (i) ξ ∧ Y => (dvol ⊗ 1Q ) for all Y ∈ Λk (C ∞ (M ))⊗Q (ii) ξ = (∗ ⊗ id)X. Proof: (i) ⇒ (ii): Let x ∈ M . Choose a coordinate neighborhood (U, x1 , x2 , ...., xn ) around x in 25

P ˆ M . Then we can write ξ(x) = |I|=n−k dxI (x)qI (x) where qI ∈ C ∞ (M )⊗Q, 1 ∞ such that {dx1 |x ,P dx2 |x , ..., dxn |x } is an ortho normal basis of Λ (C (M )) at x. ˆ Also let X(x) = |J|=k dxJ (x)QJ (x) where QJ ∈ C ∞ (M )⊗Q. Taking Y such that Y (x) = dxJ ′ (x)1Q , we have > (x)(dvol(x) ⊗ 1Q ) = dvol(x)QJ ′ (x). ′

Also (ξ ∧ Y )(x) = dvol qI ′ (x), where I is such that (dxI ′ ∧ dxJ ′ )(x) is dvol(x) ′ ′ upto a sign (note that there is unique such I corresponding to J ). Hence (ξ ∧ Y )(x) => (x)dvol(x).1Q ⇒ QJ ′ (x) = qI ′ (x) Since x was an arbitrary point in M , by definition of ∗ we conclude that ξ = (∗ ⊗ id)X. The other direction of the proof is trivial. ✷

5.3

Basics of normal bundle

We state some basic definitions and facts about normal bundle of a manifold without boundary embedded in some euclidian space. For details of the topic we refer to [35]. Let M ⊆ RN be a smooth embedded submanifold of RN . For each point x ∈ M define the space of normals to M at x to be Nx (M ) = {v ∈ RN : v ⊥ Tx (M )}. The total space M(M ) of the normal bundle is defined to be N (M ) = {(x, v) ∈ M × RN ; v ⊥ Tx (M )} with the projection π on the first coordinate. Then define Nǫ (M ) = {(x, v) ∈ N (M ); ||v|| ≤ ǫ}. With the introduced notations we have Fact: N (M ) is a manifold of dimension N . (see page no. 153 of [35]). Lemma 5.11 (i)Let BǫN −n (0) be a closed euclidean (N − n) ball of radius ǫ centered at 0. If M is a compact n-manifold without boundary embedded in some Euclidean space RN such that it has trivial normal bundle, then there exists an ǫ > 0 and a global diffeomorphism F : M × BǫN −n (0) → Nǫ (M ) ⊆ RN given by N −n X ξi (x)ui F (x, u1 , u2 , ..., uN −n ) = x + i=1

where (ξ1 (x), ..., ξN −n (x)) is an orthonormal basis of Nx (M ) for all x, and x 7→ ξi (x) is smooth ∀ i = 1, ..., (N − n). (ii) With the diffeomorphism F as above we get an algebra isomorphism πF : C ∞ (Nǫ (M )) → C ∞ (M × BǫN −n (0)) given by πF (f )(x, u1 , u2 , ..., uN −n ) = f (x + PN −n i=1 ξi (x)ui ). 26

(i) is a consequence of the tubular neighborhood lemma. For the proof see [35]. Proof of (ii) is straightforward.

We now introduce the notion of stably parallelizable manifolds. Definition 5.12 A manifold M is said to be stably parallelizable if its tangent bundle is stably trivial. We recall the following from [36]: Proposition 5.13 A manifold M is stably parallelizable if and only if it has trivial normal bundle when embedded in a Euclidean space of dimension higher than twice the dimension of M . Proof: see discussion following the Theorem (7.2) of [21]. ✷ We note that parallelizable manifolds (i.e. which has trivial tangent bundles) are in particular stably parallelizable. Moreover, given any compact Riemannian manifold M , its orthonormal frame bundle OM is parallelizable.

Part II: Smooth action and lifting to bundles 6

Smooth action of a CQG on a manifold

In this section we consider a compact manifold M possibly with boundary, but not necessarily orientable and discuss a notion of smoothness of a CQG action α of a CQG Q. Moreover we prove that smoothness automatically implies injectivitity of the reduced action, hence unitary implementability. This will follow from injectivity of the action on C ∞ (M ) which we will prove.

6.1

Definition of a smooth action

Definition 6.1 A topological action (in the sense of definition 4.4) of Q on the Fréchet algebra C ∞ (M ) is called the smooth action of Q on the manifold M . Lemma 6.2 A smooth action α of Q on M extends to a C ∗ action on C(M ) which is denoted by α again. Proof: It follows from the fact that C ∞ (M ) is stable under taking square roots of positive invertible elements. ✷ ˆ Lemma 6.3 Given a C ∗ action α : C(M ) → C(M )⊗Q, α(C ∞ (M )) ⊂ C ∞ (M, Q) ∞ ∞ if and only if (id ⊗ φ)(α(C (M ))) ⊂ C (M ) for all bounded linear functionals φ on Q. 27

Proof: Only if part ⇒ See discussion in Subsection 4.2. If part ⇒ We prove it when M is an open subset of Rn with compact closure. Then for a general compact manifold we can prove the result by going to coordinate neighborhoods. Let (u1 , ..., un ) be the standard coordinate system of Rn and let ei be the standard basis for Rn (i.e ei = (0, .., 1, .., 0), where 1 is in the i-th ∂ place), we shall show that ∂u α(f ) ∈ C ∞ (M, Q). Fix x0 ∈ M . We have to show i α(f )(x1 ,..,xi +h,..,xn )−α(f )(x1 ,..,xi ,..,xn )

0 0 0 0 0 0 is Cauchy as h → 0. For that ω(x0 ; h) := h ′ ′ that let h, h be small enough such that x0 +hei and x0 +h ei both belongs to M (this is possible since we have assumed that M is an open subset of Rn ). Then denoting the ith coordinate of x0 by xi0 , we get a ξ1 = (x10 , .., xi0 + ξ1i , .., xn0 ) ′ for some ξ1i ∈ (0, h), and ξ2 = (x10 , .., xi0 + ξ2i , .., xn0 ) for some (0, h ) and a ξ = (x10 , .., xi0 + ξ i , .., xn0 ) for some ξ i ∈ (ξ1i , ξ2i ) such that ′

|φ(ω(x0 ; h) − ω(x0 ; h ))| αφ (f )(x10 , .., xi0 + h, .., xn0 ) − αφ (f )(x10 , .., xi0 , .., xn0 ) − = |[ h ′ αφ (f )(x10 , .., xi0 + h , .., xn0 ) − αφ (f )(x10 , .., xi0 , .., xn0 ) ]| h′ ∂ ∂ = | αφ (f )(ξ1 ) − αφ (f )(ξ2 )| ∂ui ∂ui ∂2 < | 2 αφ (f )|ǫ ∂ui 2

′

∂ where ǫ = min{|h|, |h |}. Now since M has compact closure, let supx∈M | ∂u 2 αφ (f )(x)| < i Mφ where Mφ is a constant depending on φ. Now applying uniform boundedness ′ principle, we get some constant C > 0 such that ||(ω(x0 ; h) − ω(x0 ; h ))|| < Cǫ ′ for all small enough h, h from which it follows that limh→0 ω(x0 ; h) exists. The proof of existence of higher derivatives will be similar and hence it is omitted.✷

Theorem 6.4 Suppose we are given a C ∗ action α of Q on M . Then following are equivalent: 1) α(C ∞ (M )) ⊂ C ∞ (M, Q) and Sp α(C ∞ (M ))(1 ⊗ Q) = C ∞ (M, Q). 2) α is smooth. 3) (id ⊗ φ)α(C ∞ (M )) ⊂ C ∞ (M ) for all state φ on Q, and there is a Fréchet dense subalgebra A of C ∞ (M ) over which α is algebraic. Proof: (1)⇒ (2): Observe that it is enough to show that α is Fréchet continuous. Let fn → f in Fréchet topology of C ∞ (M ) and α(fn ) → ξ in Fréchet topology of C ∞ (M, Q). Then fn → f in norm topology of C(M ). So by the C ∗ continuity of α, 28

α(fn ) → α(f ). Similarly, α(fn ) → ξ in the norm topology of C(M, Q). So α(f ) = ξ and by the closed graph theorem α is Fréchet continuous. (2)⇒ (3): Follows from the Proposition 4.7. (3)⇒ (1): From Lemma 6.3, it follows that α(C ∞ (M )) ⊂ C ∞ (M, Q). The density condition follows from densities of A and A ⊗ Q0 in C ∞ (M ) and C ∞ (M, Q) respectively.✷ We end this subsection with an interesting fact which will be used later. For this, we need to recall that C ∞ (M ) is a nuclear locally convex space and hence so is any quotient by closed ideals. Lemma 6.5 If Q has a faithful smooth action on C ∞ (M ), where M is compact manifold, then for every fixed x ∈ M there is a well-defined extension of the ∞ counit map ǫ to the subalgebra Q∞ x := {αr (f )(x) : f ∈ C (M )} satisfying ǫ(α(f )(x)) = f (x), where αr is the reduced action discussed earlier. Proof: Replacing Q by Qr we can assume without loss of generality that Q has faithful Haar state and α = αr . In this case Q will have bounded antipode κ. Let αx : C ∞ (M ) → Q∞ x be map αx (f ) = α(f )(x). It is clearly continuous w.r.t. the Frechet topology of C ∞ (M ) and hence the kernel say Ix is a closed ideal, so that the quotient which is isomorphic to Q∞ x is a nuclear space. Let us consider Q∞ with this topology and then by nuclearity, the projective tensor products x with Q (viewed as a seperable Banach space, where separabiloty follows from the fact that Q faithfully acts on the seperable C ∗ algebra C(M )) coincide, and ∞ˆ ˆ will be denoted by Q∞ x ⊗Q. Thus, the multiplication map m : Qx ⊗Q → Q is ∞ˆ indeed continuous. Now, observe that Qx ⊗Q is isomorphic as a Frechet algebra ˆ by the ideal Ker(αx ⊗ id) = Ix ⊗Q. ˆ with the quotient of C ∞ (M )⊗Q Moreover, ˆ it follows from the relation ∆ ◦ α = (α ⊗ id) ◦ α that ∆ maps Ix to Ix ⊗Q, and in fact it is the restriction of the Frechet-continuous map α⊗id there, hence induces ∞ˆ ∼ ∞ ∼ ∞ ˆ ˆ a continuous map from Q∞ x = C (M )/Ix to Qx ⊗Q = (C (M )⊗Q)/(Ix ⊗Q). ∞ˆ ∞ Thus, the composite map m ◦ (id ⊗ κ) ◦ ∆ : Qx → Qx ⊗Q is continuous and this coincides with ǫ(·)1Q on the Frechet-dense subalgebra of Q∞ x spanned by elements of the form α(f )(x), with f varying in a Frechet-dense subalgebra of C ∞ (M ) on which the action is algebraic. This completes the proof of the lemma.✷ Remark 6.6 It is clear that ǫ extends to the ∗-algebra generated by Q∞ x and Q0 and the extension is still a ∗-homomorphism. This follows from the facts that (i) ǫ is ∗-homomorphism on Q0 , (ii) f 7→ ǫ(αr (f )(x)) = f T (x) is continuous with respect to the Frechet topology of C ∞ (M ), and (iii) Q∞ Q0 is dense in x Q∞ x because it contains the elements of the form αr (f )(x) for f varying in a Frechet-dense ∗-algebra on which α is algebraic (so that αr (f )(x) ∈ Q0 ).

29

Corollary 6.7 For any smooth action α on C ∞ (M ), the conditions of Theorem 4.9 are satisfied. Proof: Replacing Q by the Woronowicz subalgebra generated by {α(f )(x), f ∈ C(M ), x ∈ M } we may assume that α is faithful. If αr (f ) = 0 for f ∈ C ∞ (M ) then by Lemma 6.5 applying the extended ǫ we conclude f = 0. Now, consider any positive Borel measure µ of full support on M , with φµ be the positive functional obtained by integration w.r.t µ. Let ψ := (φµ ⊗ h)◦ αr be the positive functional which is clearly αr -invariant and faithful on C ∞ (M ), i.e. ψ(f ) = 0, f ∈ C ∞ (M ) and f nonnegative implies f = 0. But then by Riesz RRepresentation Theorem there is a positive Borel measure ν such that ψ(f ) = M f dν. It follows that ν has full support, hence ψ is faithful also on C(M ). Indeed, for any nonempty open subset U of M there is a nonzero positive f ∈ C ∞ (M ), with 0 ≤ f ≤ 1, and supportR of f is contained in U . By faithfulness of ψ on C ∞ (M ) we get 0 < ψ(f ) = U f dν ≤ ν(U ).✷

6.2

Defining dα for a smooth action α

Let α : C ∞ (M ) → C ∞ (M, Q) be a smooth action and set dα(df ) := (d⊗id)α(f ) for all f ∈ C ∞ (M ). Theorem 6.8 dα extends to a well defined continuous map from Ω1 (C ∞ (M )) ¯ satisfying dα(df ) = (d ⊗ id)α(f ), if and only if to Ω1 (C ∞ (M ))⊗Q (ν ⊗ id)α(f )α(g) = α(g)(ν ⊗ id)α(f )

(2)

for all f, g ∈ C ∞ (M ) and all smooth vector fields ν on M . Proof: Only if part ⇒ We have dα(df.g) = (d ⊗ id)α(f ).α(g), dα(g.df ) = α(g).(d ⊗ id)α(f ). But df.g = g.df in Ω1 (C ∞ (M )), which gives (d ⊗ id)α(f ).α(g) = α(g).(d ⊗ id)α(f ), ∀f, g ∈ C ∞ (M ). Observe that as ν is a smooth vector field, ν is a Fréchet continuous map from C ∞ (M ) to C ∞ (M ). Thus it is enough to prove (2) for f, g belonging to the Fréchet dense subalgebra A as in Theorem 6.4 . Let α(f ) = f(0) ⊗ f(1) and α(g) = g(0) ⊗ g(1) (Sweedler’s notation). Let x ∈ M and (U, x1 , .., xn ) be a coordinate neighbourhood around x. Pn ∂f (x)f(1) g(1) dxi |x . Then [(d ⊗ id)α(f )α(g)](x) = i=1 g(0) (x) ∂x(0) i So [(d ⊗ id)α(f )α(g)](x) = [α(g)(d ⊗ id)α(f ))](x) ∂f(0) ∂f(0) (x)f(1) g(1) = g(0) (x) (x)g(1) f(1) ⇒ g(0) (x) ∂xi ∂xi

(3)

for all i = 1, ..., n Pn ∂ Now let ai ∈ C ∞ (M ) for i = 1, ..., n such that ν(x) = i=1 ai (x) ∂x |x for all i 30

x ∈ U. So

=

[(ν ⊗ id)α(f )α(g)](x) n X ∂f(0) ai (x) (x)g(0) (x)f(1) g(1) ∂xi i=1

=

[α(g)(ν ⊗ id)α(f )](x) n X ∂f(0) ai (x) (x)g(0) (x)g(1) f(1) ∂xi i=1

and

Hence by (3) [α(g)(ν ⊗ id)α(f )](x) = [(ν ⊗ id)α(f )α(g)](x) for all x ∈ M i.e. [α(g)(ν ⊗ id)α(f )] = [(ν ⊗ id)α(f )α(g)] for all f, g ∈ A. Proof of the if part ⇒ This needs a number of intermediate lemmas. Let x ∈ M and (U, x1 , ..., xn ) be a coordinate neighbourhood around it. Choose ∂ smooth vector fields νi ’s on M which are ∂x on U . i So [α(g)(νi ⊗ id)α(f )](x) = ∂f(0) ∂xi (x)g(0) (x)f(1) g(1) .

∂f(0) ∂xi (x)g(0) (x)g(1) f(1)

and [(νi ⊗ id)α(f )α(g)](x) =

Hence by the assumption X ∂f(0) i

∂xi

(x)g(0) (x)g(1) f(1) dxi |x =

X ∂f(0) i

∂xi

(x)g(0) (x)f(1) g(1) dxi |x

⇒ [(d ⊗ id)α(f )α(g)](x) = [α(g)(d ⊗ id)α(f )](x)

Since x is arbritary, we conclude that [α(g)(d ⊗ id)α(f )] = [(d ⊗ id)α(f )α(g)] for all f, g ∈ A. So by Fréchet continuity of d and α we can prove the result for f, g ∈ C ∞ (M ).✷ We use the commutativity to deduce the following: Lemma 6.9 For F ∈ C ∞ (Rn ) and g1 , g2 , .., gn ∈ C ∞ (M ) (d ⊗ id)α(F (g1 , ..., gn )) =

n X

α(∂i F (g1 , ..., gn ))(d ⊗ id)α(gi ),

(4)

i=1

where ∂i F denotes the partial derivative of F with respect to the ith coordinate of Rn . Proof: As {(g1 (x) . . . gn (x))|x ∈ M } is a compact subset of Rn , for F ∈ C ∞ (Rn ), we get a sequence of polynomials Pm in Rn such that Pm (g1 , ..., gn ) converges to 31

F (g1 , ..., gn ) in the Fréchet topology of C ∞ (M ). We see that for Pm , (d ⊗ id)α(Pm (g1 , ..gn )) = =

(d ⊗ id)Pm (α(g1 , ..., gn )) n X α(∂i Pm (g1 , ..., gn ))(d ⊗ id)α(gi ), i=1

using (d⊗id)α(f )α(g) = α(g)(d⊗id)α(f ) as well as the Leibnitz rule for (d⊗id). The lemma now follows from Fréchet continuity of α and (d ⊗ id).✷ Lemma 6.10 Let U be a coordinate neighborhood. Also let g1 , g2 , ..., gn ∈ C ∞ (M ) be such that (g1 |U , . . . gn |U ) gives a local coordinate system on U . Then (d ⊗ id)α(f ) =

n X

α(∂gj f )(d ⊗ id)α(gj ),

j=1

for all f ∈ C ∞ (M ) supported in U . Proof: Let F ∈ C ∞ (Rn ) → R be a smooth function such that f (m) = F (g1 (m), ...., gn (m)) ∀m ∈ U. Choose χ ∈ C ∞ (M ) with χ ≡ 1 on K = supp(f ) and supp(χ) ⊂ U . Then χf = f as χ ≡ 1 on K. Hence χF (g1 , ..., gn ) = f (χF = χf = f on U, χF = 0 outside U ). Also χ2 F (g1 , ..., gn ) = χF (g1 , ..., gn ), since on K, χ2 = χ = 1 and outside K, χ2 F (g1 , ..., gn ) = χF (g1 , ..., gn ) = 0. Let T := α(χ) and ′ S := α(F (g1 , ..., gn )). Also denote (d⊗id)α(F (g1 , ..., gn )) by S and (d⊗id)α(χ) ′ by T . ′ ′ ′ ′ So we have T 2 S = T S and by (2) we have T T = T T and S S = SS . T 2S

′

= =

α(χ2 )(d ⊗ id)α(F (g1 , ..., gn )) n X 2 α(∂i F (g1 , ..., gn ))(d ⊗ id)α(gi ) (by (4)) α(χ ) i=1

= =

α(χ) α(χ)

n X

i=1 n X

α(χ∂i F (g1 , ..., gn ))(d ⊗ id)α(gi ) α(∂gi f )(d ⊗ id)α(gi ) (as supp(∂gi f ) ⊂ K).

i=1

32

(5)

TS

′

= =

α(χ)(d ⊗ id)α(F (g1 , ..., gn )) n X α(χ∂i F (g1 , ..., gn ))(d ⊗ id)α(gi ) i=1

=

n X

α(χ2 ∂i F (g1 , ..., gn ))(d ⊗ id)α(gi )

i=1

=

α(χ)

n X

α(∂gi f )(d ⊗ id)α(gi )

(6)

′

(7)

i=1

Combining (5) and (6) we get T 2S = T S

′

Now T 2S = T S ⇒ (d ⊗ id)(T 2 S) = (d ⊗ id)T S ′

′

′

′

′

′

⇒ 2T T S + T 2 S = T S + T S(by Leibnitz rule and T T = T T ) ′

′

⇒ 2T T S = T S (by(7)) ⇒ 2α(χ)(d ⊗ id)α(χ)α(F (g1 , ..., gn )) = (d ⊗ id)α(χ)α(F (g1 , ..., gn )) ⇒ 2α(χ2 )(d ⊗ id)α(χ)α(F (g1 , ..., gn )) = α(χ)(d ⊗ id)α(χ)α(F (g1 , ..., gn )) ⇒ 2(d ⊗ id)α(χ)α(f ) = (d ⊗ id)α(χ)α(f )( using the assumption and χ2 F = f ) ⇒ (d ⊗ id)α(χ)α(f ) = 0

(8)

So (d ⊗ id)α(f ) =

(d ⊗ id)α(χf )

= =

(d ⊗ id)α(χ)α(f ) + α(χ)(d ⊗ id)α(f ) α(χ)(d ⊗ id)α(f )( by (8))

= =

α(χ)(d ⊗ id)α(χF (g1 , ..., gn )) α(χ)(d ⊗ id)α(χ)α(F (g1 , ..., gn )) + α(χ2 )(d ⊗ id)α(F (g1 , ..., gn ))

=

(d ⊗ id)α(χ)α(f ) + α(χ2 )(d ⊗ id)α(F (g1 , ..., gn ))(Again by assumption) n X α(∂i F (g1 , ..., gn ))(d ⊗ id)α(gi )(by (4) and (8)) α(χ2 )

=

i=1

=

n X

α(χ2 ∂i F (g1 , ..., gn ))(d ⊗ id)α(gi )

i=1

=

n X

α(∂gi f )(d ⊗ id)α(gi )

i=1

✷ Now to complete the proof of the theorem, we want to first define a bimodule 33

morphism β extending dα locally, i.e. we define βU (ω) for any coordinate neighborhood U and any smooth 1-form ω supported in U as follows: Choose C ∞ functions g1 . . . gn as before such that they Pngive a local coordinate system on U and ω has the unique expression ω = j=1 φj dgj . Then define Pn βU (ω) := j=1 α(φj )(d ⊗ id)α(gj ). We should verify the following: Claim: βU is independent of the choice of the coordinate functions (g1 , . . . , gn ), i.e. set of coordinate functions on U with ω = Pn if (h1 , . . . , hn ) is another such ∞ ψ dh for some ψ ’s in C (M ), then j j j Pj=1 Pn n j=1 α(φj )(d ⊗ id)α(gj ) = j=1 α(ψj )(d ⊗ id)α(hj ). proof of the claim: Let χ be a smooth function which is 1 on the support of ω and 0 outside U . We have F1 , . . . , Fn ∈ C ∞ (RN ) such that gj = Fj (h1 , . . . , hn ) for all j = 1, . . . , n on U . Then χg Pjn = χFj (h1 , . . . , hn ) for all j = 1, . . . , n . Hence dgj = P k=1 ∂hk (Fj (h1 , . . . , hn ))dhk on U . That is ωP= j,k χφj ∂hk (Fj (h1 , . . . , hn ))dhk . So ψk = j χφj ∂hk (Fj (h1 , . . . , hn )). Also, note that, as χ ≡ 1 on the support of φj for all j, we must have φj ∂hk (χ) ≡ 0, so χφj ∂hk (Fj (h1 , . . . , hn )) = χφj ∂hk (χFj (h1 , . . . , hn )). Thus X

α(ψk )(d ⊗ id)α(hk )

k

=

X

α(χφj ∂hk (Fj (h1 , . . . , hn )))(d ⊗ id)α(hk )

k,j

=

X

α(φj )α(∂hk (χFj (h1 , . . . , hn )))(d ⊗ id)α(hk )

k,j

=

X

α(φj )(d ⊗ id)α(χFj (h1 , . . . , hn )) (by Lemma 6.9)

j

=

X

α(φj )(d ⊗ id)α(χgj )

j

=

X

α(φj )(d ⊗ id)α(gj )

j

Where the last step follows from Leibnitz rule and the fact that

=

α(φj )(d ⊗ id)(α(χ)) X α(φj )α(∂hk (χ))(d ⊗ id)(α(hk )) k

=

X

α(φj ∂hk (χ))(d ⊗ id)(α(hk ))

k

=

0 (using φj ∂hk (χ)) ≡ 0),

34

which proves the claim. Hence the definition is indeed independent of choice of coordinate system. Then for any two coordinate neighborhoods U and V , βU (ω) = βV (ω) for any ω supported in U ∩ V . It also follows from the definition and Lemma 6.10 that βU is a C ∞ (M ) bimodule morphism and βU (df ) = (d ⊗ id)α(f ) for all f ∈ C ∞ (M ) supported in U . Now we define β globally as follows: Choose (and fix) a smooth partition of unity {χ1 , . . . , χl } subordinate to a cover {U1 , . . . , Ul } of the manifold M such that each Ui is a coordinate neighborhood. Define β by: l X βUi (χi ω), β(ω) := i=1

for any smooth one form ω. Then for any f ∈ C ∞ (M ), β(df )

=

l X

βUi (χi df )

i=1

=

l X

βUi (d(χi f ) − f dχi )

i=1

=

l X

[(d ⊗ id)α(χi f ) − α(f )(d ⊗ id)α(χi )]

i=1

=

l X

α(χi )(d ⊗ id)α(f ) (by Leibnitz rule)

i=1

=

(d ⊗ id)α(f )

This completes the proof of the Theorem 6.8. ✷

7 7.1

Action which preserves a Riemannian inner product Definition and its implication

Definition 7.1 Suppose that we are given a possibly degenerate positive semidefinite inner product > on Ω1 (C ∞ (M )) which is faithful on Ω1 (A) for a Frechet-dense unital ∗-algebra A of C ∞ (M ) where M is a compact smooth manifold as discussed earlier. We call a smooth action α on M to preserve > on A if >= α(>) 35

(9)

for all f, g ∈ A, and where we have also used > to denote the positive semi-definite sesquilinear form on A ⊗ Q0 given by >=> q ∗ q ′ . In case the semi-definite is strictly positive definite and comes from a Riemannian structure, we call the smooth action Riemannian inner product preserving, or simply inner product preserving. Moreover it is easy to see, by Fréchet continuity of the maps d and α, the equation (9) with f, g varying in any Frechet dense ∗-subalgebra of C ∞ (M ) is equivalent to having it for all f, g ∈ C ∞ (M ). Let us now fix throughout this section as well as Section 8, a compact smooth manifold M and a faithful smooth action α of Q on it. Let us also choose the canonical Frechet dense unital subalgebra A := C ∞ (M )0 . Then by Proposition 4.7, α is algebraic and total over A and A is in fact maximal such subspace. Moreover, recall that the action satisfies any of the equivalent conditions (a)-(c) of Lemma 4.9 and let µ be a faithful, positive Borel measure µ on C(M ) given by (a) of Lemma 4.9 such that the functional say τ ≡ φµ obtained by integrating w.r.t. µ is α-invariant, i.e. Z Z α(f )(x)dµ(x) = f (x)dµ(x)1Q (10) for all f ∈ C(M ). Lemma 7.2 Assume that we are given a possibly degenerate semi-Riemannian structure on M with the associated semi-inner product > on Ω1 (A) which is preserved by α. Then there is an α equivariant (algebraic) unitary representation dα on Ω1 (A) satisfying dα(df ) = (d ⊗ id)α(f ) for all f ∈ A. Proof: P P fi dgi , for fi , gi ∈ A we define dα(ω) := For ω = i α(ai )(d ⊗ id)α(bi ). iP Also let η = i fi′ dgi′ , where fi′ , gi′ ∈ A, then >= α(>). Then it is a well defined α equivariant bimodule morphism. The coassociativity condition follows from that of α. Moreover, as Sp α(A)(1⊗Q0 ) = A ⊗ Q0 , we have Spdα(Ω1 (A))(1 ⊗ Q0 ) = Ω1 (A) ⊗ Q0 . ✷ Remark 7.3 It follows from the maximality of A that >∈ A for ω, ω ′ ∈ Ω1 (A). Indeed, let B be the subspace spanned by elements of the form > with ω, ω ′ ∈ Ω1 (A). As dα maps Ω1 (A) into Ω1 (A) ⊗ Q0 and we also have α(>) =>∈ B ⊗ Q0 , it is clear that α is algebraic on B, hence B ⊆ A. Lemma 7.4 Under the hypothesis of Lemma 7.2, dα(k) : Ωk (A) → Ωk (A) ⊗ Q0 is an α equivariant unitary representation for all k = 1, . . . , n. Proof: As α is inner product preserving, by the Theorem 7.2, we see that dα is an α-equivariant unitary representation on the bimodule Ω1 (A).For 2 ≤ k ≤ n, 36

take E1 = Ωk−1 (A), E2 = Ω1 (A), Γ1 = dα(k−1) ,Γ2 = dα,B = C = D = A0 , αB = αC = αD = α and apply Lemma 4.15 to get the desired dα(k) .✷ ✷ Now we want to show that dα(k) actually descends to the quotient module Λk (A) of Ωk (A). By the Lemma 7.4 dα(k) extends to a well defined bimodule ′ morphism from Ωk (A) to Ωk (A) ⊗ Q0 such that >= α(>) for all ω, ω ∈ Ωk (A) and Spdα(Ωk (A))(1⊗Q0 ) = Ωk (A)⊗Q0 . Also as ′ dα(k) is inner product preserving, by maximality of A, we have >∈ A ′ for ω, ω ∈ Ωk (A). Remark 7.5 Let ω ∈ Ωk−1 (A). Then by construction it is easy to see that dα(k) (dω) = (d ⊗ id)dα(k−1) (ω) for ω ∈ Ωk−1 (A). Now Recall that by Lemma 5.7, Ωk (A) = Λk (A) ⊕ JkA and JkA is a complemented submodule of Ωk (A). We have the following: Lemma 7.6 Under the hypothesis of Lemma 7.2, dα(k) leaves JkA invariant. Proof: Let ω ∈ Ωk−1 (A) such that ω = 0. Then dω ∈ JkA . We have by Remark 7.5, dα(k) (dω) = (d ⊗ id)dα(k−1) (ω). But by α-equivariance of dα(k−1) , we get >= α >= 0. Hence dα(k) leaves JkA invariant.✷ Lemma 7.7 Under the hypothesis of Lemma 7.2, dα(k) : Λk (A) → Λk (A)⊗Q0 , is α equivariant and Sp dα(k) (Λk (A))(1 ⊗ Q0 ) = Λk (A) ⊗ Q0 . Proof: ′ ′ We have for ω, ω ∈ Ωk (A), >∈ A. Using the positive faithful αinvariant functional τ mentioned in the beginning of this subsection, we define a scalar valued inner product on Ωk (A) by ′

′

< ω, ω >:= τ (>), ′

for all ω, ω ∈ Ωk (A). We denote the Hilbert space obtained as the completion of A bimodule Ωk (A) with respect to this inner product by H. Also we denote the closed subspace obtained as the completion of the submodule JkA with respect to this inner product inside the Hilbert space H by F and we denote the orthogonal projection onto this subspace by p. ′ For e, e ∈ Ωk (A), ′

< dα(k) (e), dα(k) (e ) > ′

= (τ ⊗ id) > ′

= (τ ⊗ id)α(>).1Q (by α equivariance of dα(k) ) ′

= τ (>).1Q (byαinvariance ofτ ) ′

= < e, e > 1Q 37

Hence for any h ∈ H, we can define U (h) := limn→∞ dα(en ) where en is a sequence from Ωk (A) converging to H in the Hilbert space sense and the right ¯ hand side limit is taken in the Hilbert C ∗ module H⊗Q. Then we have < ′ ′ U (h), U (h ) >=< h, h > .1Q . The fact that Sp U (H)Q is dense in the Hilbert ¯ follows from the fact that Sp dα(k) (Ωk (A))(1⊗Q0 ) = Ωk (A)⊗ C ∗ module H⊗Q Q0 . Hence U is a unitary representation of the CQG Q on the Hilbert space H. Then by Proposition 6.2 of [24], U leaves both pH and p⊥ H invariant. Let P be the orthogonal projection onto the complemented submodule JkA . Claim p⊥ H ∩ Ωk (A) = P ⊥ Ωk (A) = Λk (A). Proof of the claim: Let e ∈ p⊥ H ∩ Ωk (A). Then < e, P e >= 0, since P e ∈ JkA ∈ F . That implies τ (>) = 0 ⇒ τ (>) = 0 ⇒ >= 0(since τ is f aithf ul on A) ⇒ P e = 0. Hence e ∈ P ⊥ Ωk (A). Conversely suppose f ∈ P ⊥ Ωk (A). Then < f, P e >= τ (>) = 0. But since P Ωk (A) is dense in pH, f ∈ p⊥ H. This completes the proof of the claim. As U agrees with dα(k) on p⊥ H ∩ Ωk (A), dα(k) leaves both JkA and JkA⊥ ¯ 0 . So there exists ei ∈ Ωk (A) and qi ∈ Q0 such that invariant. Let ξ ∈ JkA ⊗Q Pk echet module Ωk (A) ⊗ Q0 . Now for i=1 dα(k) (ei )(1 ⊗ qi ) = ξ in the Hilbert Fr´ any e ∈ Ωk (A), (P ⊗ id)dα(k) (e) =

(P ⊗ id)dα(k) (P e + P ⊥ e)

= =

(P ⊗ id)dα(k) (P e)(as dα(k) (P ⊥ e) ∈ P ⊥ Ωk (A) ⊗ Q0 ) dα(k) (P e)

So l X dα(k) (ei )(1 ⊗ qi )) (P ⊗ id)( i=1

=

l X

dα(k) (P ei )(1 ⊗ qi )

i=1

Hence Sp dα(k) (JkA )(1 ⊗ Q0 ) = JkA ⊗ Q0 . Similarly considering the projection P ⊥ we can conclude that Sp dα(k) (Λk (A))(1 ⊗ Q0 ) = Λk (A) ⊗ Q0 . The α equivariance follows from that of dα(k) on Ωk (A).✷ Here we also make the convention dα0 ≡ α. As dα is a well defined bimodule morphism, by Theorem 6.8, α(f )(x) commute among themselves for different f ’s and also commute with ((φ⊗id)α(g))(x)’s where f, g ∈ C ∞ (M ) and φ is any 38

smooth vector field. For x ∈ M let us denote by Qx the unital C ∗ -subalgebra of Q generated by elements of the forms α(f )(x), ((φ ⊗ id)α(g))(x), where f, φ are as before. Using the lift of dα(2) to Λ2 (C ∞ (M )), we can show more. Indeed, we now claim that actually ((φ ⊗ id)α(g))(x)’s commute among themselves too, for different choices of φ and g. In other words: Lemma 7.8 Under the hypothesis of Lemma 7.2, each Qx is commutative. Proof: The proof is very similar to the proof of Proposition (4) of [26] for the case ˜ )(= q = 1. The statement of the lemma is clearly equivalent to proving dα(f ˜ (d⊗id)(α(f ))) and dα(g) commute for f, g ∈ A. For x ∈ M , choose smooth oneforms {ω1 , . . . , ωn } such that they form a basis of T ∗ M at x. Let FP i (x), Gi (x), i = ˜ )(x) = 1, . . . , n be elements of Q (actually in Qx ) such that dα(f i ωi (x)Fi (x), P ˜ dα(g) = i ωi (x)Gi (x). Now dα(2) leaves invariant the submodules of syms metric and antisymmetric tensor product of Λ1 (A), thus in particular, Cij = s a a s a Cji , Cij = −Cji for all i, j, where Cij and Cij denote the Q-valued coefficient of wi (x) ⊗ wj (x) in the expression of dα(2) (df ⊗ dg + dg ⊗ df )|x and dα(2) (df ⊗ dg − dg ⊗ df )|x respectively. By a simple calculation using these relations, we get the commutativity of Fi (x), Gj (x) for all i, j. ✷ We also have the following observation which follows from the constructions of dα(k) ’s and the definition of Qx . ′

′

Lemma 7.9 For every k ≥ 0, x ∈ M and ω, ω ∈ Λk (A), we have > (x) ∈ Qx . We end this subsection with the observation that if the given semi-definite form is actually a Riemannian inner product, then all the above results can be stated and proved for C ∞ (M ) replacing A. More precisely, we have: Theorem 7.10 If α is inner product preserving for a Riemannian structure then there is an α equivariant unitary representation dα on Ω1 (C ∞ (M )) satisfying dα(df ) = (d ⊗ id)α(f ) for all f ∈ C ∞ (M ). Proof: The proof is immediate from Lemma 7.10. Clearly, the inner product preserving condition extends to all f, g ∈ C ∞ (M ) by Frechet continuity of the maps involved and the density of A. It also follows from density of Ω1 (A) in the ¯ that Hilbert module Ω1 (C ∞ (M )) and density of Ω1 (A) ⊗ Q0 in Ω1 (C ∞ (M ))⊗Q ¯ Sp dα(Ω1 (C ∞ (M )))(1 ⊗ Q) = Ω1 (C ∞ (M ))⊗Q. ✷ We get the following from Lemma 7.4. Lemma 7.11 If α is Riemannian inner product preserving, then dα(k) : Ωk (C ∞ (M )) → ¯ is an α equivariant unitary representation for all k = 1, . . . , n. Ωk (C ∞ (M ))⊗Q Corollary 7.12 If α is a Riemannian inner product preserving action, then the restriction of the α-equivariant representation dα(k) of Q on the Hilbert module Ωk (C ∞ (M )) onto the closed submodule Λk (C ∞ (M )) is again an α-equivariant representation. 39

Proof: Follows from the densities of Λk (A) and Q0 in the Hilbert bimodule Λk (C ∞ (M )) and the C ∗ algebra Q respectively and the Lemma 7.7.✷

7.2

The averaging trick

Fix, as in the previous subsection before a compact, Riemannian manifold M (not necessarily orientable), a faithful smooth action α of a CQG Q and the maximal Fréchet dense subalgebra A of C ∞ (M ) over which the action α is algebraic and also the α-invariant faithful positive functional τ . In this subsection our aim is to show we can equip M with a new Riemannian structure with respect to which the action becomes inner product preserving. Note that apriori we do not assume the action to be isometric. Our strategy will be to start with the Laplacian of some Riemannian structure on M and then using the Haar state of Q somehow ‘average’ it to get a semi-Riemannian structure (faithful at least on A) for which the action becomes inner product preserving, which will imply a lot of nice things including equivariant lifts to bimodules of Higher order forms and cmutativity of Qx considered in the previous subsection. Then we will use these properties to deduce that the semi-Riemannian structure is acually Riemannian. Theorem 7.13 The manifold M has a Riemannian structure such that α is inner product preserving. For the proof we need a number of lemmas. We also claim that we can assume without loss of generality that the antipode is norm-bounded on Q and the Haar state h is faithful. To justify this claim, observe that as α is algebraic on A, the corresponding acion of the reduced CQG Qr given by (id ⊗ πr ) ◦ α (where πr denotes the quotient morphism from Q onto Qr ) coincides with α on A. For this reason, it is enough to show that the reduced action αr is inner product preserving for some Riemannian structure on A. In other words, we can replace Q by Qr in this proof, and as we already know that Qr must be a Kac algebra, antipode is norm-bounded on it. This proves the claim. Now choose a Riemannian structure on M and let T be the corresponding Laplacian which satisfies the conditions (i),(ii),(iii) of Proposition 5.6. We want to employ an averaging trick to get another linear map T satisfying similiar conditions so that the Riemannian structure corresponding to this new T will be preserved by α. Let H = L2 (M, µ), the GNS space of the α-invariant τ and let U be the unitary representation of Q on H which implements α, i.e. ˜ (π(f ) ⊗ 1)U ˜ ∗ = (π ⊗ id)(α(f )), where π denotes the faithful representation of U C(M ) on H which sends f to the multiplication operator Mf , i.e. Mf g = f g. However, we’ll often write simply f for Mf or π(f ) unless there is a chance of confusion.

40

Lemma 7.14 Let C be a densely defined linear map on H such that A ⊆ D(C) ˜ ∗ (C ⊗ I)U ˜ and C mas A into C ∞ (M ). Then we have: (i) The domain of U ∞ ˆ contains A ⊗ Q0 and it maps A ⊗ Q0 into C (M )⊗Q. ˜ ∗ (C ⊗ I)U ˜ , Mf ⊗ 1] = U ˜ ∗ [C ⊗ I, (π ⊗ id)(α(f ))]U˜ on A ⊗ Q0 for all f ∈ A. (ii) [U Proof: ˜ ∗ (f ⊗ q) = (id ⊗ κ)(α(f ))(1 ⊗ q) for f ∈ A, q ∈ Q0 . As κ We observe that U ˆ is stronger than the is norm-bounded on Q and norm-topology on C(M )⊗Q Hilbert module topology of H⊗Q, we get the above equality for all f ∈ C(M ). Now, for ξ ∈ A, q ∈ Q0 , we observe ˜ ∗ (C ⊗ I)U ˜ (ξ ⊗ q) U =

ˆ (id ⊗ κ)(α(C(ξ(0) ))(1 ⊗ ξ(1) q) ∈ C ∞ (M )⊗Q,

where α(ξ) = ξ(0) ⊗ ξ(1) in the Sweedler notation. The proof of (ii) is straightforward and hence omitted.✷ We now define T on H by Lemma 7.14 as follows: ˜ ∗ (T ⊗)U ˜ (ξ ⊗ 1), T ξ = (id ⊗ h)(U for xi ∈ A. We claim the following: Lemma 7.15 The map T satifies the conditions (i)-(iii) of Proposition 5.6 on A. Proof: By repeated applications of Lemma 7.14 we get that the domain of T indeed contains A, it maps it into C ∞ (M ) and moreover, for f1 , f2 , g1 , g2 ∈ A, g1 g2 [[T , Mf1 ], Mf2 ] ˜ ∗ (π ⊗ id)(α(g1 ))[[T ⊗ I, (π ⊗ id)(α(f1 )), ], (π ⊗ id)(α(f2 ))](π ⊗ id)(α(g2 ))U ˜ . = (id ⊗ h) U From this we can easily get condition (ii) of Proposition 5.6. To verify (i), we use Sweedler notation and get for f, g, ξ ∈ A the following: [[T , Mf ], Mg ] = = =

˜ ∗ ([[T, Mf ], Mg ]ξ(0) ⊗ f(1) g(1) ξ(1) (id ⊗ h) U (0) (0) ∗ ˜ kT (f(0) , g(0) )ξ(0) ⊗ f(1) g(1) ξ(1) (id ⊗ h) U

(id ⊗ h) (id ⊗ κ)(F (ξ(00) ⊗ ξ(01) ))(1 ⊗ f(1) g(1) ξ(1) ) ,

41

where F = α(kT (f(0) , g(0) )). Using the facts that h is tracial, κ is antihomomorphism, h = h ◦ κ, κ2 = id we write the above as (id ⊗ h) (1 ⊗ κ(f(1) g(1) ξ(1) )F (ξ(00) ⊗ ξ(01) ) = (id ⊗ h) ξ(00) ⊗ ξ(01) κ(ξ(1) κ(g(1) f(1) )F = θξ, where θ = (id ⊗ h)((1 ⊗ κ(g(1) f(1) )F ), and also where we have used the identity ξ(00) ⊗ ξ(01) κ(ξ(1) = ξ(0) ǫ(ξ(1) = ξ. Let us also note that (f, g) 7→ (α(f ), α(g) is Frechet-continuous, hence so is (f, g) 7→ Φ(f, g) := kT (f(0) , g(0) )ξ(0) ⊗ f(1) g(1) ξ(1) . If f, g ∈ C ∞ (M ), fn , gn ∈ A0 converging to f, g respectively in the Frechet topology, then kT (fn , gn ) = ˜ ∗ Φ(fn , gn )U ˜ ) converges in norm to say kT (f, g) ∈ C(M ). This proves (id ⊗ h)(U the last statement in (i) of Proposition 5.6. To verify (iii) of Proposition 5.6, we use the faithfulness of h, hence that of the ∞ ˜∗ ˜ ˆ map X 7→ (id⊗h)( P U X U ) for X ∈ C (M )⊗Q. Thus, for f1 , . . . , fn , g1 , . . . , gn ∈ A such that ij gi gj kT (f i , fj ) = 0, we have X ij

α(gi )∗ kT (f i(0) , fj (0) ) ⊗ f i (1)fj (1) α(gj ) = 0

P .P Applying (id ⊗ ǫ) on the above, we obtain ij g i gj kT (f i , fj ) = 0, hence g df = 0 from the fact that T satisfies (iii) of Proposition 5.6. ✷ i i i Thus we get a faithful possibly non-degenerate semi-inner product in the sense of Proposition 5.6, say >T , on Ω1 (A). We claim that this is preserved by α. Lemma 7.16 α presevres >T . Proof: ˜ on A ⊗ Q0 which we This will follow from the commutativity of T ⊗ I and U prove first. Denote as usual the H⊗Q valued inner product by > and the inner products of H or H ⊗ L2 (Q, h) by < ·, · >. Also, note that h(x) =< 1, x1 > for any x ∈ Q. For ξ, η ∈ A we have < ξ, T η > ˜ (ξ ⊗ 1), (T ⊗ I)(U ˜ (η ⊗ 1)) > = h(ξ(1)

42

Using this, we have for ξ, η ∈ A, q, q ′ ∈ Q, ˜ (ξ ⊗ q), (T ⊗ I)U ˜ (η ⊗ q ′ ) >> q ∗ ξ(1) η(1) q ′

=

∗ ∗ < ξ(00) , T < ξ(00) , T η(00) > q ∗ h(ξ(01) η(01) )ξ(1) η(1) q ′

=

∗ ∗ η(12) q ′ η(11) )ξ(12) < ξ(0) , T η(0) > q ∗ h(ξ(11)

=

∗ η(1) )q ′ < ξ(0) , T η(0) > q ∗ h(ξ(1)

=

< ξ, T η > q ∗ q ′

=

>,

which proves the claim that ˜ =U ˜ (T ⊗ I) (T ⊗ I)U

(11)

˜ and U ˜ ∗ leave A ⊗ Q0 invariant and are inverses to one another on A ⊗ Q0 . As U ˜ replaced by U ˜ ∗ and hence on this domain, we also have (11) with U [[(T ⊗ I), α(f )], α(g)] ˜ ([[T , Mf ], Mg ] ⊗ I)U ˜ ∗ (ξ ⊗ q) = U = α(kT (f, g))(ξ ⊗ q) for f, g, ξ ∈ A, q ∈ Q0 . That is, ˜ ˜ >T = kT (f (0) , g(0) )f(1) ∗ g(1) = α(kT (f , g)) = α(>T ), which proves the lemma. ✷ As a corollary of Lemma 7.16 above, we get that for any fixed x, the algebra Qx generated by (d ⊗ id)(α(f ))(x), α(g)(x), with f, g varying over A, and hence C ∞ (M ), is commutative and hence in particular for f, g ∈ A, kT (f(0) , g(0) )f(1) g(1) T belongs to C C ∞ (M ) ⊗ Q0 , where C is the commutative ∗-subalgebra of ˆ consisting of all F such that F (x) ∈ Qx ∀x. C ∞ (M )⊗Q T Lemma 7.17 Let ξ ∈ A, F ∈ C (C ∞ (M ) ⊗ Q0 ) and MF be the operator ˜ (ξ ⊗ 1) = Mβ (ξ ⊗ 1) for ˜ ∗ MF U of left multiplication by F on H ⊗ Q. Then U ∗ ∞ ˜ ˆ β = U (F ) ∈ C (M )⊗Q. Proof: ˜ (ξ ⊗ ˜ (ξ ⊗ 1) = α(ξ) ∈ C, so commutes with F . Thus, MF U Observe that U ˜ (Mξ ⊗ 1)U ˜ ∗ (F ). Applying U ˜ ∗ on the left, we obtain 1) = F α(ξ) = α(ξ)F = U ˜ (ξ ⊗ 1) = (ξ ⊗ 1)β = β(ξ ⊗ 1) = Mβ (ξ ⊗ 1), where β = U ˜ ∗ (F ) ∈ ˜ ∗ MF U U ∞ ∞ C (M, Q) as F ∈ C (M ) ⊗ Q0 .✷

43

˜ ∗ (F )(p) = 0 implies ǫ(F (p)) = Lemma 7.18 For F ∈ C ∞ (M ) ⊗ Q0 , p ∈ M , U 0. Proof: P many terms and fi ∈ Let F = i fi ⊗ qi where the sum is over finitely ˜ ∗ (F )(p) = P (id ⊗ κ)(α(fi ))(p)qi . ApplyC ∞ (M ), qi ∈ Q0 . We also have 0P =U i ing κ (which is bounded) we get κ(q )α(f )(p) = 0. Using the extension of ǫ i i i P ǫ(q )f (p) = 0, i.e. ǫ(F (p)) = 0, as ǫ ◦ κ = ǫ on Q0 .✷ on Q∞ we have i i p i

Proof of Theorem 7.13: We have already got a semi-Riemannian structure on M which is preserved by α. To complete the proof we need to show that the condition (iv) of Proposition 5.6 holds for the kernel kT corresponding to T . To this end, consider P f1 , . . . , fn , g1 , . . . , gn ∈ A such that g (p)g (p)k j ij i T (f i , fj )(p) = 0for some P ∗ kT (f i(0) , fj (0) ) ⊗ f i (1)fj (1) α(gj ) ∈ p ∈ M . Now, taking F = ij α(gi ) T ∞ ˜ ∗F U ˜ )(p) = C (C (M ) ⊗ Q0 ) we get by Lemma 7.17 that 0 = (id ⊗ h)(U ∗ ∗ ˜ ˜ h(U (F )(p)) and β = U (F ) is clearly positive element of Q. By faithfulness ˜ ∗ (F )(p) = 0, and hence F (p) = 0 by Lemma 7.18. Applying ǫ on of h we get U P P F (p) we get ij gi (p)gj (p)kT (f i , fj )(p) = 0, which implies i gi (p)dfi (p) = 0 as T satisfies condition (iv) of Proposition 5.6. ✷

8

Lifting inner product preserving action to cotangent and frame bundles

Let α be a smooth action on C ∞ (M ) as in the previous section, hence it also satisfies the condition (10) with a measure µ, say, and suppose furthermore that we have obtaind a Riemanian inner product for which it is inner product preserving. For a fixed ǫ > 0 let Eǫk (M ) := {(m, ω) : m ∈ M, ω = (ω1 , . . . , ωk ), ωi ∈ T ∗ M, < ωi , ωj >= δij ǫ}. This is a subbundle of the direct sum of k copies of cotangent bundle, say E = T ∗ (M ) ⊕ ... ⊕ T ∗ (M ). The fibre at each point is K = {v ≡ (v1 , ..., vk ) : | {z } k−copies

vi ∈ Rn ∀i and < vi , vj >= δij ǫ}. Also let E be the of the C ∞ (M ) bimodule of sections of the complexified bundle EC . The total space Eǫk (M ) is again a smooth, compact manifold. For a trivializing neighborhood U of ′ ′ M for the bundle π −1 (U ) ∼ = U × K. On U choose {ω1 , ..., ωn } such that ′ ′ {ω1 (m), ..., ωn (m)} forms an orthonormal basis for Tǫ∗ (M ) for all m ∈ U . Then if we denote the points of the total space of the bundle by e, we de′ (U,ω ′ ) fine tij (e) ≡ tij (e) :=< ωi (m), ωj (m) > for all 1 ≤ i ≤ k and 1 ≤ j ≤ n, where π(e) = m and π is the projection map of the bundle. The definition of 44

tij ’s do depend on the choice of local trivialization given by U and ωi′ ’s but we will prefer not to make this dependence notationally explicit, unless absolutely necessary. Note also that for k = n and ǫ = 1, the fibre is diffeomorphic to On (R) and the bundle is the orthonormal frame bundle. The fibre K admits a natural action by On (R), given by (v1 , . . . , vk ) 7→ (T v1 , . . . , T vk ) whete T ∈ On (R). Indeed, Eǫk is an associated bundle of a principal On (R)-bundle. There is a canonical On (R)-invariant Riemannian metric as well as corresponding Riemannian volume measure of full support on K. For example, namely k = 1 i.e. K = S n−1 we can choose this Riemannian metric as the standard On (R)-invarant metric on S n−1 induced from its embedding in Rn , Let us fix any such metric on K and denote it by >0 and the corresponding measure by µ0 Now, we have the product Riemannian metric on π −1 (U ) ∼ = U × K for any locally trivializing neighbourhood. As the structure group of the bundle is reduced to On (R) and >0 and µ0 are On (R)invariant, we can verify easily that the above product metric as well as product measure do not depend on the choice of locally trivializing neighbourhoods or choice of local orthonormal bases of T ∗ M giving the local Bundle charts, hence gives a well-defined Riemannian structure (and the corresponding positive Borel measure of full support) on the total space of the bundle. Let B = C(Eǫk (M )), B ∞ = C ∞ (Eǫk (M )), BU = Cc (π −1 (U )) ⊂ C(Eǫk (M ))), ∞ BU = Cc∞ (π −1 (U )) ⊂ C ∞ (Eǫk (M )), where Cc (V ) (Cc∞ (V )) denotes continuous (smooth respectively) compactly supported functions on the manifold V . Consider the ∗-algebra PU (PU∞ respectively) consisting of the functions of the form n X fˆl Pl (tij : i = 1, ..., k; j = 1, ..., n), l=1

where N ≥ 1, fˆl ≡ fl ◦ π ∈ Cc (U × K)(Cc∞ (U × K)) respectively for fl ∈ Cc (U )(Cc∞ (U )) and Pl denotes the polynomial in nk variables. It is easy to see ∞ that PU (PU∞ respectively) is dense in BU (BU respectively) in norm (Fréchet respectively) topology. We shall also use the notation Fˆ for F ∈ C(M, Q) to denote the element F ◦ π of C(Eǫk (M ), Q). We want to construct a smooth action η of Q on Eǫk (M ). For that first observe that there is a natural C ∞ (M ) valued inner product E on E. This is given by > (m) := Pk ∞ k i=1 < ωi (m), ηi (m) >. Define θ : E → C (Eǫ (M )) by θ(ξ)(e) := hhω, ξiiE (π(e)),

where e = (m, ω). It is easy to check that it is a module map where we view C ∞ (M ) as a subalgbera of C ∞ (Eǫk (M )) in the obvious way, by sending f to fˆ. So, by Lemma 4.17 we get an algebra homomorphism θˆ from S(E) to C ∞ (Eǫk (M )). In fact it is a ∗ algebra homomorphism. We have Lemma 8.1 (i) θ : E → C ∞ (Eǫk (M )) is continuous in the respective Fréchet topologies. (ii) Image of θˆ is dense in C ∞ (Eǫk (M )) in its Frechet topology. 45

Proof: (i) Follows from the Fréchet continuity of . (ii) Choose f ∈ C ∞ (Eǫk (M )). Fix an open neighbourhood U of M and ′ ′ a local frame {ω1 , ..., ωn } such that π −1 (U ) ∼ = U × K. Then Cc∞ (π −1 (U )) ∼ = ∞ ∞ ∞ C (K, Cc (U )). The algebra PU (as mentioned earlier) is dense in the Fréchet ′ topology of Cc∞ (π −1 (U )). Choose η ij = (0, ..., ωj , ..., 0), where ωj′ is at the ith place, 1 ≤ i ≤ k.. Then for (m, ω) ∈ π −1 (U ), by definition θ(η ij )(m, ω) = ˆ ij ) = gtij , where g ∈ C ∞ (π −1 (U )). tij (m, ω). As θˆ is a module map, we have θ(gη c ˆ In this way we get an arbritary element of PU∞ in the image of θ. Now let f ∈ C ∞ (Eǫk (M )). Eǫk (M ) being a compact manifold, there is a finite cover say {Ui }i=1,...,l for some integer l. Choose a partition of unity {χi } subordinate to this cover. Then f χi ∈ C ∞ (Ui ) and hence by the previous ar(n) ˆ i (n)) → χi f in the gument there is some sequence si ∈ S(E) such that θ(s Pl P (n) l ˆ Fréchet topology. Clearly θ( i=1 si ) → i=1 f χi = f , thereby completing the proof of (ii).✷ Now let E0 be a Fréchet dense submodule of E such that θˆ is algebraic over S(E0 ). Combining (i) of Lemma 4.18 and Lemma 8.1, we conclude that ∞ k ˆ ˆ Lemma 8.2 θ(S(E 0 )) is dense in C (Eǫ (M )) and hence (θ ⊗ id)(S(E0 ) ⊗ Q0 ) ∞ k is dense in C (Eǫ (M ), Q).

Now, consider an inner product preserving smooth action α on C ∞ (M ). Recall that dα is an α-equivariant unitary representation on Ω1 (C ∞ (M )). Taking the direct sum representation we get another α equivariant unitary representation of Q on E (by Lemma 4.14). We denote this representation by Γ. By Lemma 4.13, we get a Fréchet dense submodule E0 of E on which Γ is algebraic and Sp Γ(E0 )(1 ⊗ Q0 ) = (E0 ⊗ Q0 ). Applying (ii) Lemma 4.18 (as Qπ(e) is commutative) we can conclude that the representation Γ lifts to an equivariant coassociative sym ˆ map Γsym (n) of Q on each E(n) , so we get a coassociative map Γ : S(E0 ) → S(E0 ) ⊗ Q0 . Also define Ψ : E → C ∞ (Eǫk (M ), Q). ¯

Ψ(ξ)(m, ω) :=>E ⊗Q , ¯ ¯ valued inner product. Consider the subwhere E ⊗Q denotes the E ⊗Q ∞ algebra C of (C (Eǫk (M ), Q) consisting of those F such that F (m, ω) ∈ Qm for all (m, ω) ∈ Eǫk (M ). This is a commutative algebra. Hence again applying ˆ from S(E) → C. We also have on S(E0 ) the following Lemma 4.17 we get a lift Ψ

ˆ = Ψ. ˆ (θˆ ⊗ id)Γ

(12)

Now we are ready to define a smooth action η of Q on Eǫk (M ). Fix a trivializing ′ ′ neighborhood U and a local orthonormal sections ω1 , ..., ωn as before. Then ∞ k define Tij ∈ C (Eǫ (M ), Q) for i = 1, ..., k and j = 1, ..., n as ′

Tij (m, ω) :=> (m), 46

where denotes the C ∞ (M, Q) valued inner product as before and ω = (ω1 , ..., ωk ). It follows that Tij ∈ C for all 1 ≤ i ≤ k and 1 ≤ j ≤ n. Moreover, we have Lemma 8.3 For any smooth real-valued function χ supported in U , we have (i) X [ il ǫ, [ Tij Tlj = α(χ)δ (13) α(χ) j

for all i, l = 1, . . . , k, where δil are the Kronecker’s delta. (ii) Let V be any other trivializing neighbourhood with local orthonormal sections ′′ {ωj′′ , j = 1, . . . , n} and P let′′ tij ’s be the corresponding ‘matrix coefficients’. If we ′ write dα(ωj )(m) = l ωk (m)Fjk (m), for m ∈ V , for some Fjk (m) ∈ Qm , we have X X Fji Fjl = α(χ)δil , (14) Fij Flj = α(χ) α(χ) j

j

for all i, l = 1, . . . , n, Proof: Fix e = (m, (ω1 , . . . , ωk ) ∈ Eǫk , m = π(e). Let γ be a character (multiplicative linear funtional) on the commutative C ∗ algebra Qm and uj := (id ⊗ ∗ γ)(dα(ωj′ )(m)) ∈ Tm M . By a simple calculation using the fact that dα is innerproduct preserving and ω ′ j ’s form an orthonomal basis of T ∗ M at every point in the support of χ, we obtain γ(α(χ)(m))2 < uj , ul > = γ (> (m)) = γ α(χ)2 (m)α(>)(m) γ(α(χ2 )(m))δjl

=

Thus, in case γ(α(χ(m))) is nonzero, u1 , . . . , un is an orthonormal basis of ∗ Tm M and by multiplicativity of γ and self-adjointness of Tij ’s it is easy to see that γ(Tij (e)) =< ωi (π(e)), uj > Thus, we have 

γ

X

=

j



Tij (e)Tlj (e)

X

< ωi (π(e)), uj >< ωl (π(e)), uj >

j

= < ωi (π(e)), ωl (π(e)) >= ǫδil ,

47

[ = \2 X = δil α(χ) \2 ǫ, where X = P Tij Tlj which implies α(χ)X which proves α(χ) j [ [ is self adjoint, thereby proving (i). δij α(χ)X, as α(χ) For (ii), using the definition of Tij ’s in terms of t′′ij and Fij ’s and also the P relations (13) and j t′′ij t′′lj = δil ǫ, we get on π −1 (V ) the following: α(χ)(π(e))

X

t′′ir (e)t′′lr (e) = α(χ)(π(e))

r

X

t′′ir (e)t′′ls (e)qrs (π(e)),

(15)

r≤s

P where qrs = j Fjr Fjs Here we have also used the commutativity of the elements involved. However, there is a homoemorphism between fibres of the bundle Eǫk with On (R) whic identifies tij ’s with the function on On (R) which takes a matrix to its ij-th entry. Thus, from the defining relations of On (R) we conclude that {t′′ir t′′ls , (i, r) 6= (j, s), r ≤ s} are linearly independent. Comparing coefficients of t′′ir t′′ls in (15) we get α(χ)qrs = δrs α(χ). Now, given any character γ on the commutative C ∗ algebra Qπ(e) such that γ(α(χ)(π(e))) is nonzero, this means ((γ(Fjr (π(e))))) is an orthogonal matrix, so we also have P P γ j Frj Fsj = α(χ)δrs . j Frj (π(e))Fsj (π(e)) = δrs 1, i.e. α(χ) This completes proof of (ii).✷ Recall the algebra PU . Denote by EU the C ∞ (M ) bimodule of smooth sections of EC supported in U and let SU0 be the subalgebra of S(EU ) ⊂ S(E) PN consisting of elements of the form l=1 fˆl Pl (η ij : i = 1, . . . , k, j = 1, . . . , n) (N ≥ 1), where Pl , fl are as in the definition of PU , η ij are as in the proof of ˆ 0 ) = PU . It is easily seen that Lemma 8.1. Clearly, θ(S U N N X X ˆ fˆl Pl (η ij ; i = 1, . . . , k, j = 1, . . . , n)) = θ( (fˆl )Pl (tij ; i = 1, . . . , k, j = 1, . . . , n), i=1

i=1

N N X X ˆ fˆl Pl (η ij ; i = 1, . . . , k, j = 1, . . . , n))) = Ψ(( (fˆl )Pl (Tij ; i = 1, . . . , k, j = 1, . . . , n), i=1

i=1

ˆ ˆ Lemma 8.4 ||Ψ(g)|| ≤ ||θ(g)|| for all g ∈ SU0 . Proof: Indeed, as Tij ’s are self-adjoint elements of the commutative C ∗ algebra C, it is clear that the map is a ∗-homomorphism provided it is well-defined, which immediately follows once we verify the norm estimate. To this end, Note the natural identification of the C ∗ -algebra BU with C(K, Cc (U )) ⊂ C(K, C(M )), so that we have kψk = supv∈K kψ ( v)k, for ψ ∈ C(K × M ) ≡ C(K, C(M )), where ψ v (x) = ψ(v, x) for v ∈ K). A similar fact holds if we replace C(M ) by ˆ C(M )⊗Q. We also denote by α♯ the C ∗ -homomorphism sending ψ ∈ C(K ×M )

48

ˆ to α♯ (ψ) in C(K, C(M )⊗Q) given by, α♯ (ψ)(v) = α(ψ v ), v ∈ K. Clearly, ♯ kα (ψ)k ≤ kψk. PN Now, consider g ∈ SU0 of the form l=1 fˆl Pl (ξij ; i, j = 1, . . . n), and assume that L is a compact subset of U in which supports of all fl ’s are contained. Let χ ∈ C ∞ (U ) be such that 0 ≤ χ ≤ 1 and χ|L ≡ 1. Fix e ∈ Eǫk (M ) and let γ be a character (multiplicative functional) of the commutative C ∗ algebra Qπ(e) . By [ Lemma 8.3, we see that if γ(α(χ)(e)) is nonzero, then τ = (τ1 , . . . , τk ) belongs to K, where τi = (τi1 , . . . , τin ), with τij = γ(Tij (e)). On the other hand, we have (using the fact that fl = χfl for each l): ˆ γ(Ψ(g)(e)) [ ˆ = γ(α(χ)(e))γ( Ψ(g)(e)) =

N X

γ(α(fl )(π(e)))Pl (τij ; i = 1, . . . k, j = 1, ..., n)

l=1

= γ

N X

!

α(fl )(π(e))Pl (Tij ; i = 1, . . . k, j = 1, ..., n) .

l=1

[ ˆ Thus, if γ(α(χ)(e)) is zero, Pγ(Ψ(g)(e)) = 0, andwhen it is nonzero, we obtain N ˆ ˆ |γ(Ψ(g)(e))| ≤ supv∈K k α(fl )(π(e))Pl (v) k, which is equal to kα♯ (θ(g))k ≤ l=1

ˆ kθ(g)k. ✷

ˆ satisfyLemma 8.5 There exists a C ∗ -action η : C(Eǫk (M )) → C(Eǫk (M ))⊗Q ˆ ˆ ing η(θ(g)) = Ψ(g). Proof: ˆ ˆ From the previous Lemma 8.4 and density of SU0 in S(EU ), we get ||Ψ(g)|| ≤ θ(g) for g ∈ S(EU ). Now subordinate to a finite cover {U1 , U2 , ..., Ul }, choose a smooth partition of unity {χ1 , χ2 , ..., χl }. Then for g ∈ S(E), we have ˆ ||Ψ(g)|| =

||

l X

ˆ i g)|| Ψ(χ

i=1

≤

l X

ˆ i g)|| ||Ψ(χ

i=1

=

l X

ˆ i g)|| ||θ(χ

i=1

≤

ˆ l||θ(g)||

So this allows us to get a bounded C ∗ homomorphism from the norm closure of ˆ ˆ θ(S(E)) = C(Eǫk (M )) to C(Eǫk (M ))⊗Q.✷

49

Lemma 8.6 η is a smooth action of Q. Moreover, η preserves the faithful positive functional on C(Eǫk ) coming from the positive Borel measure ν on Eǫk which is uniquely determined by µ × µ0 on π −1 (U ) ∼ U × K for any trivializing neighbourhood U , where µ is the faithful Borel measure on M preserved by α and µ0 denotes the On (R)-invariant measure on K. Also, it is inner product preserving for the product Riemannian structure on the total space of Eǫk Proof: As Tij ’s are in C ∞ (Eǫk (M ), Q), we get η(C ∞ (Eǫk (M )) ⊂ (C ∞ (Eǫk (M )), Q). ˆ ˆ ˆ ˆ Moreover, observe that η(θ(S(E 0 ))) = (θ ⊗ id)Γ(S(E0 )). So Sp η(θ(S(E0 )))(1 ⊗ ˆ ˆ Q0 ) = Sp (θˆ ⊗ id)Γ(S(E ))(1 ⊗ Q ). But Sp Γ(S(E ))(1 ⊗ Q ) = (S(E 0 0 0 0 0 ) ⊗ Q0 ). ∞ k ˆ ˆ Hence Frechet closure of Sp η(θ(S(E0 )))(1 ⊗ Q0 ) is C (Eǫ (M ))⊗Q. Now, we prove the claim that η preserves the product measure to be denoted by ν. It is enough to prove this for the subalgebra PU0 for any locally trivializing neighbourhood U considered before. Choose and fix any such U and also a χ as in Lemma 8.4. Consider a function F ∈ PU0 of the form F (e) = f (π(e))P (tij (e), i = 1, . . . , k; j = 1, . . . , n) as in Lemma 8.4, where P is some polynomial and f has a compact support within U , so that χ = 1 on the support of f . Now, fix another trivializing neighbourhood VR as in Lemma 8.3 and the corresponding t′′ij , Fjr etc. Note that the integral π−1 (V ) Gdν = R intm∈V dµ(m) K Gm dµ0 , where Gm is the restriction of G to the fibre at m which is homeomorphic to K. In particular, Z Z Z η(F )dν = α(f )(m)α(χ)(m) P (Tij (e), i = 1, . . . , k; j = 1, . . . , n) dµ0 . π −1 (V )

e∈π −1 {m}

m∈V

Now, for any character γ on Qm , either γ(α(χ)(m)) is zero or we have by (14) of Lemma 8.3 that the elements ((τij := γ(Fjl (m)))jl ∈ On (R) and γ(Tij (e)) = P ′′ ′′ ′′ ′′ l τjl til (e). In other words, writing Ti = (Ti1 , . . . , Tin ), ti = (ti1 , . . . , tin ), we see that (γ(T1 (e)), . . . , γ(Tk (e))) is the element of K obtained from the element R (t′′1 , . . . , t′′k ) by the µ0 -preserving action of On (R). Hence π−1 {m} γ (α(χ(m))P (Tij (e), i, j)) dµ0 = R γ(α(χ(m)))P (t′′ij (e), i, j)dµ0 . From this, we get the same relation withπ −1 {m} out γ, i.e. for all m ∈ V, Z Z α(χ(m))P (Tij (e), i, j)dµ0 = α(χ(m))P (t′′ij (e), i, j)dµ0 . π −1 {m}

π −1 {m}

R

Now, π−1 {m} P (t′′ij (e), i, j)dµ0 does not depend on m and is equal to C = R ψ(y)dµ0 (y), where ψ : K →R R given by ψ(y ≡ R(y1 , . . . , yk )) = P (yij , i, j). K R This gives, π−1 (V ) η(F )dµ = C V α(f )α(χ)dµ = C V α(f ). As this is true for R every locally trivializing V we get by a partition of unity argument E k η(F )dν = R R Rǫ R R C M α(f )dµ = C( M f dµ)1Q = ( F dν)1Q , as F dν is clearly C M f µ. The proof of the fact that it is also inner-product preserving is very similar to the proof of product-measure preservation, and omitted. ✷

50

S ∗ Let Tǫ∗ (M ) := {(m, v) : v ∈ Tm (M ), ||v|| ≤ ǫ} = 0≤r≤ǫ Er1 for some fixed positive number ǫ. We can lift the smooth action α on the manifold M to a smooth action on the total space of Tǫ∗ (M ) following the proof of lifting to Eǫk almost verbatim. Moreover, the lift will be inner-product preserving if the ′ original action is so. We denote the lifted smooth action on Tǫ∗ (M ) by η to distinguish it from the lift η on Eǫk for each fixed ǫ. Tǫ∗ (M ) is a compact 2n dimensional manifold. Note that π −1 (U ) ∼ = U ×K, where K is an n-dimensional closed ball of radius ǫ. Moreover Tǫ∗ (M ) is orientable with the following natural orientation. At the point (m, ω) ∈ π −1 (U ) and any choice ω1 , ..., ωn as before, dvol(m, ω) ∈ Λ2n (C ∞ (Tǫ∗ (M ))) is given by (ω1 ∧ ω2 ∧ ... ∧ ωn ∧ dt1 ∧ ... ∧ dtn )(m, ω). It can be seen to be independent of choice of ω1 , ..., ωn and also it is non zero everywhere. Henceforth, we shall consider Tǫ∗ (M ) oriented with the globally defined non vanishing dvol as the choice of orientation. ′

Lemma 8.7 The lifted action η is also orientation preserving in the sense ′ dη(2n) (dvol) = dvol ⊗ 1Q . Proof: For m ∈ M , choose a trivializing neighborhood around m and one forms ω1 , ..., ωn such that {ω1 (x), ..., ωn (x)} forms an orthonormal basis for Tx∗ (M ) for all x ∈ U . Then there are Q-valued functions P fij ’s for 1 ≤ i, j ≤ n such that fij (m) ∈ Qm for all m ∈ M and dα(ωi )(m) = j fij (m)ωj (m). Choose and fix some smooth non-negative function χ supported in U . By the commutativity of Qm , we get dα(n) (ω1 ∧ ... ∧ ωn )(m) X fij ωj )(m) = ∧nj=1 ( =

X

j

(sgn σ)f1σ(1) (m)f2σ(2) (m)...fnσ(n) (m)(ω1 ∧ ... ∧ ωn )(m)

σ∈Sn

= ∆(m)(ω1 ∧ ... ∧ ωn )(m). where ∆(m) = det ((fij (m))). Also, we have ′

=

dηn (dt1 ∧ ... ∧ dtn )(m, ω) X fij (m)dtj (m, ω)) ∧( j

=

∆(m)(dt1 ∧ ... ∧ dtn )(m, ω)

′

Hence dη2n (ω1 ∧ ... ∧ ωn ∧ dt1 ∧ ... ∧ dtn )(m, ω) = ∆(m)2 (ω1 ∧ ... ∧ ωn ∧ dt1 ∧ ... ∧ dtn )(m, ω). Now note that α(χ)(m)2 > (m) = δij α(χ)2 (m) 51

as wi ’s are orthonormal on the support of χ. Moreover, each fij (m) is self adjoint. Choosing any ∗-character γ on the commutative C ∗ algebra Qm as before, we see that either γ(α(χ)(m)) = 0 or ((γ(fij (m)))) is in On (R) and its determinant γ(∆(m)) is 1 or −1. Thus α(χ)∆2 = α(χ), which implies ′ dη2n (χdvol) = α(χ)(dvol ⊗ 1),

and hence by a partition of unity argument we complete the proof that η ′ is orientation-preserving.✷

Part III: Main results about about non existence of genuine quantum group actions 9 9.1

Isometric actions of CQG Definition of isometric action

Recall from the discussion at the end of the subsection about quantum isometry group (subsection 4.4) the definition of QISOL for a spectral triple satisfying certain regularity conditions. In particular, all classical spectral triples, i.e. those coming from the Dirac operator on the spinor bundle of a compact Riemannian spin manifold, do satisfy such conditions and hence QISOL is defined for them. In fact it easily follows from [15] that one can go beyond spin manifolds and define (and prove existence of) such a quantum isometry group for any compact Riemannian manifold M (without boundary) as the universal object in the category of CQG Q with a faithful action α on C(M ) such that (id ⊗ φ)α(C ∞ (M )) ⊂ C ∞ (M ) for all state φ and commutes with the Hodge Laplacian (to be called the L2 Laplacian) L2 = −d∗ d restricted to L2 (M, dvol). We shall denote the universal object in this category by QISOL (M ) in this paper. It is proved in (Theorem 3.8 of [11]) that QISOL (M ) ∼ = QISOI+ (d + d∗ ) where now d is viewed as a map on the Hilbert space of forms of all orders, i.e. M k Λ (M ). the L2 closure of ⊕dim k=0 Furthermore it follows from the Sobolev theorem that (id ⊗ φ)α(C ∞ (M )) ⊂ C ∞ (M ) for all state φ. We have the following Theorem 9.1 QISOL (and hence any subobject in the category QL ) has a smooth action on C ∞ (M ). Proof: We denote the C ∗ action of QISOL on C(M ) by α. By Sobolev embedding theorem, for any state φ on QISOL , (id⊗φ)(C ∞ (M )) ⊂ C ∞ (M ). Let {eij : j = 1, ..., di } be the orthonormal eigen vectors of L forming a basis for the eigen space corresponding to the eigen vector λi . We denote Sp {eij : 1 ≤ j ≤ di , i ≥ 1} ∞ by A∞ 0 . Then this is a subalgebra of C (M ). Furthermore, it is easy to see ∞ that α is algebraic over A0 and hence total. The proof of the theorem will 52

be complete by applying Lemma 6.4, if we can show that A∞ echet dense 0 is Fr´ in C ∞ (M ) which is a consequence of Sobolev theorem. However we include a proof for the sake of completeness. The idea is similar to that of lemma 2.3 of [15]. ′ By Theorem 1.2 of [13] There are constants C and C such that ||eij ||∞ ≤ n−1 n−1 ′ C|λi | 4 and di ≤ C |λi | 2 , where n is the dimensionPof the manifold. For f ∈ C ∞ (M ) there are complex numbers fij such that ij fij eij converges to P f in L2 norm. Since f ∈ dom(Lk ) for all k ≥ 1, ij |λi |2k |fij |2 < ∞ for all k. P Choose and fix sufficiently large k such that i≥0 |λi |n−2k < ∞. P P L( ij fij eij ) = ij λi fij eij converges to L(f ) in the L2 norm. By CauchyScwartz inequality, X X ′ 1 X 1 |fij |2 |λi |2k ) 2 ( |λi |n−2k ) < ∞. |λi fij |||eij ||∞ ≤ C(C ) 2 ( ij

ij

i≥0

P P Hence L( ij fij eij ) = ij λi fij eij converges to L(f ) in the sup norm of C(M ). P Similarly we can show that Lk ( ij fij eij ) converges in the sup norm of C(M ) for any k. So A∞ echet dense in C ∞ (M ). 0 is Fr´ ✷

In other words isometric actions of a CQG are always smooth in the sense of subsection 6.1. Let us denote by L the restriction of L2 to C ∞ (M ), viewed as a Fréchet continuos operator (to be called the ’geometric Laplacian’). When M is oriented we can also write it as −(∗d)2 , where ∗ is the Hodge * operator as discussed in subsection 5.2. As C ∞ (M ) is a core for L2 , it is clear that a CQG ˆ is isometric (i.e. (A, α) is an object in QL ) if and action α : C(M ) → C(M )⊗Q only if α is smooth and commutes with L in the sense that α ◦ L = (L ⊗ 1)α. For the purpose of this paper, we need to extend the above formulation of quantum isometry group to manifolds with boundary. Choosing Dirichlet boundary condition, we take d to be the closure of the unbounded operator with domain C = {f ∈ C ∞ (M ) : f |∂M=0 }. Definition 9.2 For a compact manifold with boundary we call a smooth action ˆ to be isometric if it maps C into C ⊗Q ˆ and commutes α : C(M ) → C(M )⊗Q with L2 on C ∞ (M ). Remark 9.3 For a manifold with boundary, commutation with the geometric Laplacian L may not be sufficient to imply that α is isometric. We also recquire ˆ the condition that α(C) ⊂ C ⊗Q. We can prove the existence of QISOL as well as the smoothness of the action of QISOL as in [15]. It is a consequence of the fact that the Dirichlet Laplacian has discrete spectrum with finite dimensional n−1

eigen spaces and the estimate ||ej (f )||∞ ≤ Cλj 2 ||f ||2 of the eigen vectors of the Laplacian (see page 9 of [48]). ˆ be a smooth action (as introduced earlier) and Let α : C ∞ (M ) → C ∞ (M )⊗Q let us fix the maximal Fréchet dense subalgebra A of C ∞ (M ) over which the 53

action is algebraic i.e. α(A) ⊂ (A ⊗ Q0 ) and Sp α(A)(1 ⊗ Q0 ) = A ⊗ Q0 . Note ¯ that for f ∈ C ∞ (M ), (d ⊗ id)α(f ) ∈ Ω1 (C ∞ (M ))⊗Q Lemma 9.4 If α commutes with the geometric Laplacian L on A, then >= α >

(16)

for all f, g ∈ A and hence α is inner product preserving. Proof: > ∗ g(1) = > ⊗f(1) ∗ = [L(f(0) g(0) ) − L(f(0) )g(0) − f(0) L(g(0) )] ⊗ f(1) g(1)

On the other hand α(>) = α[L(f¯g) − L(f¯)g − f¯L(g)] ∗ g(1) ( since α commutes with L) = [L(f(0) g(0) ) − L(f(0) )g(0) − f(0) L(g(0) )] ⊗ f(1)

✷

9.2

Geometric characterization of isometric action

Our aim of this Subsection is to prove that every inner product preserving smooth action is isometric. Lemma 9.5 Let N be an m-dimensional compact, oriented, Riemannian manifold (possibly with boundary) with dvol ∈ Λm (C ∞ (N )) be a globally defined nonzero form. Moreover let η be a smooth inner product preserving action on N such that dη(m) (dvol) = dvol ⊗ 1. Then η commutes with the geometric Laplacian. Proof: First we note that as η is an inner product preserving smooth action, by 7.12 it lifts to an α-equivariant unitary representations dη(k) : Λk (C ∞ (N )) → ¯ for all k = 1, ..., m. Note that without loss of generality we can Λk (C ∞ (N ))⊗Q dvol replace dvol by 1 and assume that >= 1, since if 2

dη(m) preseves dvol, it also preserves the normalized dvol. First we claim that We have ∀ k = 1, ..., m, dη(m−k) (∗ω) ∧ β => (dvol ⊗ 1Q ) 54

(17)

¯ ∀ β ∈ Λk (C ∞ (N ))⊗Q. ′ ′ For that let β = dη(k) (ω )(1 ⊗ q ). Then dη(m−k) (∗ω) ∧ β ′

′

=

dη(m−k) (∗ω) ∧ dη(k) (ω )(1 ⊗ q )

=

η > (dvol ⊗ q ) (by Lemma 7.9)

′

′

On the other hand from unitarity of dη(k) , ′

′

> ′

′

= η > (1 ⊗ q ). P So by replacing β by finite sums of the type i dη(k) (ωi )(1 ⊗ qi ), we can show that ω ∈ Λk (C ∞ (N )) and β ∈ Sp dη(k) Λk (C ∞ (N ))(1 ⊗ Q), dη(m−k) (∗ω) ∧ β => (dvol ⊗ 1Q ). ¯ Now, since Sp dη(k) (Λk (C ∞ (N ))(1 ⊗ Q) is dense in Λk (C ∞ (N ))⊗Q, we get a k ∞ sequence βn belonging to Sp dη(k) (Λ (C (N )))(1 ⊗ Q) such that βn → β in ¯ the Hilbert module Λk (C ∞ (N ))⊗Q. But we have dη(m−k) (∗ω) ∧ βn => (dvol ⊗ 1Q ). Hence the claim follows from the continuity of and ∧ in the Hilbert ˙ ∞ (N )⊗Q. ¯ module Λ(C ✷ Now combining Lemma 5.10 and (17) we immediately conclude the following: dη(m−k) (∗ω) = (∗ ⊗ id)dη(k) (ω) f or k ≥ 0.

(18)

Now we can prove that η commutes with the geometric Laplacian of N . For φ ∈ C ∞ (N )), η(∗d ∗ dφ) = =

(∗ ⊗ id)dη(m) (d ∗ dφ) (by equation (18) with k = m) (∗d ⊗ id)dη(m−1) (∗dφ)

= =

(∗d ⊗ id)(∗ ⊗ id)dη(dφ) (again by equation (18))) (∗d ⊗ id)(∗d ⊗ id)η(φ)

=

((∗d)2 ⊗ id)η(φ).

We are now ready to prove the main result of this section.

55

ˆ be a smooth action which is also Theorem 9.6 Let α : C(M ) → C(M )⊗Q inner product preserving on a compact Riemannian manifold. Then α commutes with the geometric Laplacian L = −(∗d)2 of the manifold M . Proof: ′ We apply the Lemma 9.6 to the manifold N = Tǫ∗ (M ) and η to be the lift of α on C ∞ (Tǫ∗ (M )) constructed in Section 8. Recall the canonical volume form ′ dvol on Tǫ∗ (M ) and that dη(2n) (dvol) = dvol ⊗ 1 from lemma 8.7. Now on a trivializing neighborhood U of M , π −1 (U ) ∼ = U × K where K is the closed Euclidean n-ball of radius ǫ. It is easy to see that locally the geometric Laplacian of the manifold Tǫ∗ (M ) (say LT ) is nothing but (L ⊗ 1 + 1 ⊗ LRn ) where LRn is the geometric Laplacian of Rn and L is the geometric Laplacian of the manifold M . Note that we are given the product Riemannian structure on U × K here and also observe that dvol is unit volume form. LT (f g) = (Lf ).g + f LRn (g). ′

On f ∈ Cc∞ (U ), identify f with f ⊗1 on U ×K and observe that η (f ⊗1) = α(f ) and LT (f ⊗ 1) = L(f ). On the other hand ′

η ((L ⊗ 1 + 1 ⊗ LRn )(f )) = α(L(f )). Thus α(L(f )) = (L ⊗ 1)α(f ). Now for any f ∈ C ∞ (M ), choose a partition of unity {χi : i = 1, ..., l} subordinate to a trivializing cover U1 , ..., Ul and observe that α(L(f ))

=

l X

α(L(χi f ))

i=1

=

l X

(L ⊗ 1)α(χi f )

i=1

=

(L ⊗ 1)α(f ).

✷ Combining the above theorem and the remarks in section 4.1 we get ˆ is a smooth action and Corollary 9.7 Suppose that α : C(M ) → C(M )⊗Q ˆ where in case M has a non trivial boundary, assume also that α(C) ⊂ C ⊗Q C = {f ∈ C ∞ (M ); f |∂M = 0}. Then α is isometric if and only if α is inner product preserving. So combining the results of this and earlier subsections we get the following Theorem 9.8 A faithful C ∗ -action of a CQG on C(M ) (where M is a compact smooth manifold without boundary) is smooth if and only if it is isometric with respect to some Riemannian structure on M .

56

10

Lifting an action to the tubular neighborhood of a stably parallelizable manifold

As before let M be a compact, oriented Riemannain n-manifold without boundary. Assume furthermore that M is stably parallelizable and let M ⊂ RN be an embedding with trivial normal bundle, for sufficiently large N ≥ n. Let Q be a CQG which acts faithfully on M as in the sense mentioned earlier. Suppose b is the (co)-action which is smooth and α commutes that α : C(M ) → C(M )⊗Q with the geometric Laplacian, say L. Now as in subsection 4.2, we have the Fréchet dense spectral subalgebra A = (C ∞ (M ))0 of C ∞ (M ) over which α is algebraic and Sp (α(A)(1 ⊗ Q0 )) = A ⊗ Q0 . Now since M is a manifold with a trivial normal bundle, Recall from Lemma 5.11, the global diffeomorphism F and corresponding isomorphism πF : C ∞ (Nǫ M ) → C ∞ (M × BǫN −n (0)). Define b = σ23 ◦ (α ⊗ id) α b : A ⊗ C ∞ (BǫN −n (0)) → A ⊗ C ∞ (BǫN −n (0)) ⊗ Q0 by α

and extend α ˆ as a Fréchet continuous map. Now we have πF : C ∞ (Nǫ M ) → C ∞ (M × BǫN −n (0)), which implies that πF −1 : C ∞ (M × BǫN −n (0)) → C ∞ (Nǫ M ). Hence b → C ∞ (Nǫ M )⊗Q. b (πF −1 ⊗idQ ) : C ∞ (M × BǫN −n (0))⊗Q

So, defining

we get

b Φ := (πF −1 ⊗ id) ◦ α b ◦ πF : C ∞ (Nǫ M ) → C ∞ (Nǫ M )⊗Q,

Theorem 10.1 Φ is a smooth action of Q on C(Nǫ M ). Proof: The smoothness is straightforward to see from the definition of the action Φ.✷ Set A˜ := πF −1 (A ⊗ C ∞ (BǫN −n (0))). Then A˜ is a Fréchet dense subalgebra of ˜ C (Nǫ (M )). By construction, Φ is algebraic over A˜ and moreover, Sp Φ(A)(1⊗ ˜ Q0 ) = A ⊗ Q0 . As the normal bundle of the manifold is trivial, we can choose smoothly varying basis for normal space at each point of the manifold. Let y ∈ Nǫ (M ) and {ei (y) : i = 1, . . . , (N − n)} be an orthonormal basis for the normal space to the manifold at the point π(y) and u1 , u2 , ..., uN −n be components of U(y) := (y − π(y)) with respect to the basis {ei (y) : i = 1, . . . , (N − n)} . We introduce a coordinate system for the manifold Nǫ (M ) as follows: ∞

57

F −1

ξ×id

G : Nǫ (M ) → M × Bǫ(N −n) (0) → RN (ξ is a coordinate map for M ) : y → (π(y), U(y)) → (x1 , ...xn , u1 , ...uN −n ). Suppose that φ ∈ C ∞ (Nǫ (M )) such that φ(y) ≡ Ψ(U(y)). Then clearly = 0 which implies that

∂ ∂xi φ

dφ =

N −n X i=1

Lemma 10.2
= 0 ∂ui ∂xj

for all i, j, i = 1, ...N − n, j = 1, ...n. Proof: We have Nǫ (M ) ⊂ RN and let (y1 , y2 , ..., yN ) be the usual coordinate functions for RN . Without loss of generality let φ ∈ C ∞ (Nǫ (M )) and y ∈ Nǫ (M ) be an interior point (for points on the boundary the proof will be similar) and φ ∈ C ∞ (Nǫ M ) and y ∈ Nǫ M such that G−1 (0, ...0) = y. Let ei (y) = (e1i (y), ...eN i (y)) for all i = 1, ...N − n. Then ∂ φ)(y) = ( ∂u i

d −1 (0, ...t, ...0)) (t dt |t=0 φ(G d = dt |t=0 φ(ξ −1 (0) + tei )

=

in ith position)

PN

j ∂φ j=1 ei (y) ∂yj |y ,

where yj ’s are coordinate functions for RN . Therefore we have PN j ∂ ∂ j=1 ei ∂yj . ∂ui =

∂ |y is nothing but the vector ei = (eji (y); j = 1, . . . , N ) under the That is, ∂u i canonical identification of RN with Ty RN . As ei (y) ∈ Nπ(y) (M ) and ∂x∂ k ∈ ∂ |y , ∂x∂ k |y >= 0. ✷ Tπ(y) (M ) for every y, < ∂u i

˜ Lemma 10.3 Φ preserves inner-product on A. Proof: Consider φ, ψ ∈ A˜ of the form φ(y) = ξ ◦ π(y) and ψ(y) = η ◦ U(y), where ξ, η are smooth functions. Then Φ(φ)(y) = α(ξ)(π(y)), Φ(ψ) = ψ ⊗ 1, and moreover by Lemma 10.2, we observe that >= 0 and > = > = 0 58

Using this and Leibnitz formula we get >= Φ(>) for f1 , f2 of the form fi = (ξi ◦ π)ηi ◦ U, i = 1, 2. As a general element of A˜ is a finite sum of product functions of the form ˜ (ξ ◦ π).(η ◦ U), we get the inner product preserving property of dΦ on A.✷ Corollary 10.4 Φ satisfies the conditions of Corollary 9.7. Hence Φ commutes P ∂2 with the geometric Laplacian LRN on Nǫ M which is same as N i=1 ∂y 2 , where i

yi ’s are coordinate functions for RN .

11 11.1

Non existence of genuine CQG action The stably parallelizable case

Let {yi : i = 1, .., N } be the standard coordinates for RN . We will also use the same notation for the restrictions of yi ’s if no confusion arises. Definition 11.1 A twice continuously differentiable, complex-valued function Ψ defined on a non empty, open set Ω ⊂ RN is said to be harmonic on Ω if LRN Ψ ≡ 0, where LRN ≡

PN

∂2 i=1 ∂yi2 .

We note the following result whose proof is essentially the same as the proof of Theorem 2.2 of [8]. We, however, give the details for the sake of completeness. Lemma 11.2 Let C be a unital commutative C ∗ algebra and x1 , x2 , . . . , xN be self adjoint elements of C such that {xi xj : 1 ≤ i ≤ j ≤ N } are linearly independent and C be a unital C ∗ algebra generated by {x1 , x2 , ..., xN }. Let Q be a compact quantum group acting faithfully on C such that the action leaves the span of {x1 , x2 , ..., xN } invariant. Then Q must be commutative as a C ∗ algebra, i.e. Q ∼ = C(G) for some compact group G. Proof: Let us introduce a convenient terminology: call a finite dimensional vector space V of a commutative algebra quadratically independent if the dimension of the subspace {vw, v, w ∈ V } is equal to the square of the dimension of V . Clearly, this is equivalent to the following: for any basis {v1 , v2 , . . . , vk } of V , {vi vj , i ≤ j ≤ k} will be linearly independent, hence a basis of V ⊗sym V . From this definition we also see that any nonzero subspace of a quadratically independent space V is again quadratically independent. Let us now denote the action of Q by α and set V := Sp{xi , i = 1, . . . , N } which is quadratically independent subspace of dimension N in C. We claim that without loss of generality we can assume that there is an inner product on V for 59

which α|V isa unitary representation. Indeed, we recall from Subsection 4.2 the maximal algebraic subspace C0 which is a direct sum of ker(α) and the spectral subspace for the action. In fact, as α is ∗-preserving, we can have a similar decomposition of the real algebra C s.a. consisting of self-adjoint elements of C. Thus, we can decompose V into real subspaces V0 ⊕ V1 , where V0 ⊆ Ker(α) and V1 ⊆ C s.a. is contained in the spectral subspace for the action. In particular, α is injective on the algebra generated by V1 . Moreover, V1 must be nonzero, because otherwise Q will be 0, and thus V1 is quadratically independent. By replacing V by V1 if necessary, we can assume that V is contained in the spectral subspace for the action α, so in particular, α is a non-degenerate algebraic representation on V and moreover, both α and the Haar state (say h) of Q are faithful on the ∗-algebra generated by the elements of V . Choose some faithful positive functional φ on the unital separable C ∗ algebra C and consider the convolved functional φ = (φ ⊗ h) ◦ α which is clearly faithful on the ∗-algebra generated by {xi }’s and also Q-invariant, so that α gives a unitary representation w.r.t. the inner product say < ·, · >φ coming from φ on V . As A is commutative and xi are self-adjoint, so are xi xj for all i, j, and hence the inner product < xi , xj >φ ’s are real numbers. Thus, Gram-Schmidt orthogonalization on {x1 , ...xN } will give an orthonormal set {y1 , ..., yN } consisting of self-adjoint elements, with the same span as V = Span{x1 , ..., xN }. Replacing xi ’s by yi ’s, let us assume for the rest of the proof that {x1 , ..., xN } is an orthonormal set, there are Qij ∈ Q, i, j = 1, . . . , N such that Q = C ∗ (Qij , i, j = 1, . . . , N ) and α(xi ) =

N X

xj ⊗ Qij , ∀i = 1, . . . , N.

j=1

Since x∗i = xi for each i and α is a ∗-homomorphism, we must have Q∗ij = Qij ∀i, j = 1, 2, ..., N. The condition that xi ,xj commute ∀i, j gives Qij Qkj = Qkj Qij ∀i, j, k,

(19)

Qik Qjl + Qil Qjk = Qjk Qil + Qjl Qik . −1

∗

(20)

T

As Q = ((Qij )) ∈ MN (Q) is a unitary, Q = Q = Q := ((Qji )), since in this case entries of Q are self-adjoint elements. Clearly, the matrix Q is an N -dimensional unitary representation of Q, so Q−1 = (id ⊗ κ)(Q), where κ is the antipode map. So we obtain T κ(Qij ) = Q−1 (21) ij = Qij = Qji . Now from ( 19 ) , we have Qij Qkj = Qkj Qij . Applying κ on this equation and using the fact that κ is an antihomomorphism along with ( 21 ) , we have Qjk Qji = Qji Qjk Similarly , applying κ on ( 20 ), we get Qlj Qki + Qkj Qli = Qli Qkj + Qki Qlj ∀i, j, k, l.

60

Interchanging between k and i and also between l, j gives Qjl Qik + Qil Qjk = Qjk Qil + Qik Qjl ∀i, j, k, l.

(22)

Now, by (20 )-( 22 ) , we have [Qik , Qjl ] = [Qjl , Qik ], hence [Qik , Qjl ] = 0. Therefore the entries of the matrix Q commute among themselves. However, by faithfulness of the action of Q, it is clear that the C ∗ -subalgebra generated by entries of Q must be the same as Q, so Q is commutative.✷ Remark 11.3 In fact, for the applications we are in mind, namely for smooth actions on manifolds which are proved to be one-to-one, the arguments in the beginning of the proof leading to the decomposition V = V0 ⊕ V1 is not necessary as V0 = 0. Thus, we can actually consider the restriction of α on V to be nondegenerate in the cases of interest. However, we have chosen to state and prove the lemma in the most general form for any future applications in wider context. Remark 11.4 If the subset V = Sp{x1 , x2 , . . . , xN } contains 1, we call any action preserving V affine. Lemma 11.5 Let W be a manifold (possibly with boundary) embedded in some RN and {yi }’s for i = 1, ..., N , be the coordinate functions for RN restricted to W . If W has non empty interior in RN , then {1, yi yj , yi : 1 ≤ i, j ≤ N } are linearly independent, i.e. {1, y1 , . . . , yN } are quadratically independent. Proof: P P If possible let on W c.1 + cij yi yj + k dk yk = 0 for some cij , dk . pick an interior point y ∈ W . Then at y, we can take partial derivatives in any direction. P ∂ |y ∂y∂ j |y to c.1 + cij yi yj + dk yk = 0, we conclude that Hence applying ∂y i cij = 0 ∀ i, j. Similarly we can prove dk ’s are 0 and hence c = 0 . ✷ Lemma 11.6 Let Φ be a smooth action of a CQG on a compact subset of RN which commutes with LRN , Then Φ is affine i.e. Φ(yi ) = 1 ⊗ qi +

N X

yj ⊗ qij , f or some qij , qi ∈ Q

j=1

for all i = 1, ..., N , where yi′ s coordinates of RN .

61

Proof: As Φ commutes with the geometric Laplacian and LRN ∂y∂ j = 0 for all j, we get (LRN ⊗ id)(

∂ ∂yj LRN ,

LRN yj =

∂ ⊗ id)Φ(yi ) ∂yj

∂ ⊗ id)Φ(LRN yi ) ∂yj = 0. = (

∂ ⊗ id)Φ(yj ))(y). Note that as dΦ is an Φ-equivariant unitary Let Dij (y) = (( ∂y i representation, by Lemma 4.16 ((Dij (y)))i,j=1,...,N is unitary for all y ∈ Nǫ (M ). Pick y0 in the interior of Nǫ M (which is non empty). Then the new Q valued matrix ((Gij (y))) = ((Dij (y)))((Dij (y0 )))−1 is unitary (since Dij (y) is so). Gij (y) is unitary for all y ⇒ |ψ(Gij (y))| ≤ 1 And |ψ(Gii (y0 ))| = 1. ψ(Gii (y)) is a harmonic function on an open connected set Int(Nǫ M ) which attains its supremum at an interior point. Hence by corollary 1.9 of [2] we conclude that ψ(Gii (y)) = ψ(Gii (y0 )). ((Gij (y))) being unitary for all y, Gij = δij .1Q . Then ((Dij (y)))((Dij (y0 )))−1 = 1MN (Q) . So ((Dij (y))) = ((Dij (y0 ))) for all y ∈ Nǫ (M ). Hence Φ is affine with qij = Dij (y0 )✷

Remark 11.7 This is the only place where we have made use of the assumption that the manifold is connected. ✷ Corollary 11.8 Let M be a smooth, compact, orientable, connected, stably parallizable manifold. Then if α is a faithful smooth action of of a CQG Q. Then Q must be commutative as a C ∗ algebra i.e. Q ∼ = C(G) for some compact group. Proof: We already proved that then we can equip the manifold with a Riemannian inner product so that α will be inner product preserving. Then we can lift the action to the tubular neighborhood of the manifold M so that it is still inner product preserving. Hence the lifted action commutes with the geometric Laplacian of the tubular neighborhood which is an open subset of RN for some N and hence by applying Lemma 11.6, Lemma 11.5 and Lemma 11.2, we complete the proof.✷

11.2

The general case

Theorem 11.9 Let α be a smooth, faithful action of a CQG Q on a compact, connected smooth manifold M . Then Q must be commutative as a C ∗ algebra i.e. Q ∼ = C(G) for some compact group G. 62

Proof: As α is smooth, we can equip M with a Riemannian structure such that α will be inner product preserving. As α is inner product preserving, we can lift the action to the total space of the orthonormal frame bundle of the manifold (apply the results from Section 8 with k = n and ǫ = 1) which is again smooth. Now the total space of orthonormal frame bundle being connected, compact, orientable, parallelizable manifold, we can apply 11.8 to complete the proof.✷ Corollary 11.10 The quantum isometry group of a compact, connected, Riemannian manifold coincides with the classical isometry group of the manifold. Proof: Follows from the fact that an isometric action of a compact quantum group satisfies the conditions of the previous theorem. ✷ Remark 11.11 In [14], Etingof and Walton proved that if a finite dimensional semisimple Hopf algebra has an ‘inner faithful’ (which is basically what we call faithful, in the dual form) action on a commutative domain then the Hopf algebra must be co-commutative, i.e. group algebra. Put in the dual form, it means that no genuine (i.e. noncommutative) finite dimensional semisimple Hopf algebra can have a faithful (in our sense) co-action on a commutative domain. Their arguments crucially depend on finite dimensionality and it is not clear how to compare it with our arguments. However, finite dimensional quantum groups are indeed semisimple Hopf algebras, and it follows from our main result of this section that one can have results similar to Etingof-Walton for a large class of infinite dimensioal Hopf algebras. More precisely, it follows that there cannot be any faithful (co)-action of a CQG Hopf ∗ algebra i.e. unital Hopf ∗-algebras with faithful positive integrals, on a large class of commutative domains arising in geometry. As a simple concrete example, let F be a polynomial in n real variables such that F is regular on X = {(x1 , . . . , xn ) : F (x1 , . . . , xn ) = 0}, and assume also that X is compact and connected. Then the algebra OX generated by the coordinate functions is a commutative domain and any faithful algebraic (co)-action of a CQG Hopf ∗-algebra on it can be easily shown to extend to a smooth faithful (co)-action of the universal CQG corresponding to the Hopf algebra, hence the Hopf algebra must be commutative.

12

+ Application: QISOR for manifolds obtained by deformation of classical manifolds using 2-cocycles

In this section we cite the main results without proof from [18] about computing the quantum group of orientation preserving Riemannian isometries for a class

63

of manifolds obtained by cocycle twisting of classical manifolds. For details of such deformation we refer the reader to [25]. We briefly recall the necessary facts.

12.1

Preliminaries about cocycle twisting

Let Q be a compact quantum group. Recall from Subsection 4.1 the dense Hopf ∗-algebra Q0 spanned by the matrix coefficients of its inequivalent irreducible ˆ be the dual discrete quantum group Q. With these representations. Also let Q notations we have the following Definition 12.1 By a unitary 2-cocycle σ of a compact quantum group Q, we ˆ⊗ ˆ satisfying ˆ Q) mean a unitary element of M(Q ˆ = (σ ⊗ 1)(∆ ˆ ⊗ id)σ. (1 ⊗ σ)(id ⊗ ∆)σ ˆ We recall from [24] the multiplier Hopf ∗-algebra Qˆ0 (with the coproduct ∆ ˆ defined by ∆(ω)(a ⊗ b) := ω(ab), w ∈ Qˆ0 and a, b ∈ Q0 ). Q0 and Qˆ0 are in non degenerate pairing. We can deform the multiplier Hopf ∗-algebra Qˆ0 using ˆ⊗ ˆ The ∗ structure, algebraic structure and the antipode do not ˆ Q). σ ∈ M (Q change where the coproduct changes by −1 ˆ σ (.) = σ ∆(.)σ ˆ ∆ .

ˆ σ is again coasUsing the cocycle condition of σ, it can be easily shown that ∆ siciative. We denote the deformed multiplier Hopf ∗-algebra by Qˆ0σ . Now σ viewed as a linear functional on Q0 ⊗Q0 satisfies the cocycle condition (see page 64 of [25]) σ(b(1) , c(1) )σ(a, b(2) c(2) ) = σ(a(1) , b(1) )σ(a(2) b(2) , c), for a, b, c ∈ Q0 ( Sweedler’s notation). We can deform Q0 using σ to obtain a new Hopf ∗-algebra Qσ0 . Then Qσ0 and Qˆ0σ again form a non degenerate pairing. We twist the product of the algebra Qσ0 by the following formula: a.σ b := σ −1 (a(1) , b(1) )a(2) b(2) σ(a(3) , b(3) ), for a, b ∈ Q0 . The coproduct remains unchanged. The ∗ structure and κ gets changed by the formulae: X v −1 (a(1) )a∗(2) v(a(3) ), a∗σ := κσ (a) := U (a(1) )κ(a(2) )U −1 (a(3) ).

(see page 65 of [25]). Definition 12.2 The deformation of the CQG Q using a unitary 2-cocycle σ is defined to be the universal compact quantum group containing Qσ0 as Hopf ∗-algebra and is denoted by Qσ . 64

Let us now discuss how one gets a unitary 2-cocycle on a CQG from such a unitary 2-cocycle on its quantum subgroup. Given two CQG’s Q1 , Q2 and a surjective CQG morphism π : Q1 → Q2 which identifies Q2 as a quantum subgroup of Q1 (we shall use the notation Q2 ≤ Q1 to mean that Q2 is a quantum subgroup of Q1 ), it can be shown that π maps the Hopf ∗-algebra ′ ′ (Q1 )0 onto (Q2 )0 . By duality we get a map say π ˆ from (Q2 )0 to (Q1 )0 and it is easy to check that this indeed maps the dense multiplier Hopf ∗-algebra (Qˆ2 )0 ⊂ Qˆ2 to (Qˆ1 )0 . Indeed π ˆ lifts to a non degenerate ∗-homomorphism from M(Qˆ2 ) to M(Qˆ1 ). So given a unitary 2-cocycle σ on Q2 , we get a unitary ′ ′ ˆ Qˆ1 ). It is easy to check that σ is again a 2-cocycle σ := (ˆ π⊗π ˆ )(σ) ∈ M(Qˆ1 ⊗ ′ unitary 2-cocycle on Q1 . We shall often use the same notation for both σ and ′ σ i.e. denote σ by σ under slight abuse of notation for convenience. Also recall from Subsection 4.1 the notion of universal compact quantum groups. With the above notations, we have the following ′

Lemma 12.3 1. Qσ2 is a quantum subgroup of Qσ1 . −1 2. For a universal CQG Q with a unitary 2-cocycle σ, (Qσ )σ ∼ = Q. Let Q be a CQG having a unitary representation U on a Hilbert space H. We assume that Q has a unitary cocycle σ. We have the following (see [18]) Proposition 12.4 U is again a unitary representation of the deformed CQG Qσ . We denote the same (as a linear map) U as Uσ when viewed as a representation of the twisted CQG. Thus for a universal CQG Q we get a bijective correspondence between the sets of inequivalent irreducible representations of Q and Qσ ; given by, ˆ Qˆσ = {πσ |π ∈ Q}, ˆ denotes the set of inequivalent irreducible representation of Q. We also where Q denote by Fπσ the positive invertible matrix corresponding to πσ . More explicitly ∗

hσ (tπij ∗ tπkl ) =

1 δik Fπσ (j, l), Mdσπ

where Mdσπ = T r(Fπσ ) and hσ is the Haar state of Qσ .

12.2

Main result

We begin by describing a general scheme for twisting a spectral triple (A∞ , H, D, R) which is equivariant with respect to a CQG action by a unitary 2-cocycle on ′ the CQG. Recall from Subsection 4.4 the category QR (D). Definition 12.5 A unitary 2-cocycle for (A∞ , H, D, R) is a unitary 2-cocycle ′ on some object in the category QR (D).

65

Now assume that the spectral triple (A∞ , H, D, R) has a unitary 2 cocycle σ ′ i.e. for some object (Q, V ) in the category QR (D), Q has a unitary 2-cocycle σ. Then H = ⊕k≥1 Hk , where each Hk is an eigen space for D. Since U commutes with D, U also preserves each Hk . So we have the following decomposition of Hk . Hk = ⊕π∈Ik ⊂Qˆ Cdπ ⊗ Cmπ,k , where mπ,k is the multiplicity of the irreducible representation π on Hk , and Ik ˆ In [18] it is proved that R must be of the following is some finite subset of Q. form R|Hk = ⊕π∈Ik Fπ ⊗ Tπ,k , for some Tπ,k ∈ B(Cmπ,k ). We define Rσ := ⊕π∈Ik ,k Fπσ ⊗ Tπ,k . Now recall from subsection 4.2, the von Neumann algebraic action adV˜ of ′′ Q on (A∞ ) and let B be the corresponding SOT dense spectral subalgebra of ∞ ′′ (A ) . We have a SOT dense subspace N (spanned by the eigen vectors of D) of H such that V maps N into N ⊗ Q0 . As adV˜ is algebraic on B, we can define ρσ (b)ξ = σ −1 (b1 , ξ1 )b0 ξ0 , f or b ∈ B, ξ ∈ N , where adV˜ (b) = b0 ⊗ b1 and V (ξ) = ξ0 ⊗ ξ1 (Sweedler’s notation). Then it can be shown that ρσ (b) can be extended as a bounded operator on whole of H. Let B σ = ρσ (B). The following facts are proved in [18]. ˆ Proposition 12.6 1) ρσ Pπ is SOT continuous for all π ∈ Q. 2) There is a SOT dense * subalgebra A0 ⊂ B such that [D, a] ⊂ B(H) for all a ∈ A0 . ′′ ′′ 3) (ρσ (A0 )) = (B σ ) . 4) ((A0 )σ , H, D) is a spectral triple where (A0 )σ = ρσ (A0 ). With the above notations and assumptions, + ^+σ (D ) ∼ ^+ σ ∼ Theorem 12.7 QISO and hence QISOR σ (Dσ ) = σ = (QISOR (D)) R + σ (QISOR (D)) .

Remark 12.8 One can also prove similar result for the quantum isometry group based on Laplacian, whenever it makes sense. In other words, QISOL (Dσ ) ∼ = (QISOL (D))σ if the spectral triples under consideration satisfy the assumptions for defining QISOL . Combining this with the result that the result that there are no genuine quantum isometries of a compact connected classical manifold, we get the following:

66

Corollary 12.9 (i) Let M be a compact connected Riemannian manifold and σ be a unitary 2-cocycle on C(ISO(M )). Then QISOL of the twisted noncommutative manifold, say Mσ , is isomorphic with C(ISO(M ))σ . (ii) If furthermore M has a spin structure and D is the corresponding Dirac operator and σ is a unitary 2-cocycle on C(ISO+ (M )) then QISOI+σ (Dσ ) ∼ = C(ISO+ (M ))σ . One can go beyond quantum isometries and prove rigidity results for more general quantum group symmetries on cocycle-deformed C ∗ algebras. As a concrete case, consider any compact connected smooth manifold M such that there is a smooth Tn -action on it. As a special case of cocycle deformation, one can construct Rieffel-deformation Mθ for any skew-symmetric n× matrix θ. There is a natural fanily of seminorms and a Frechet-algebra Mθ∞ which is also a norm-dense subalgebra of Mθ . See [30] for a detailed discussion on this. There is also a natural action on Tn on Mθ . Theorem 12.10 Suppose that a CQG Q acts faithfully on Mθ such that the action α restricts to a topological action on the Frechet algebra Mθ∞ mentioned above, and moreover, there is there is a surjective CQG morphism π from Q to C(Tn ) such that (id ⊗ π) ◦ α coincides with the canonical Tn -action on Mθ . Then Q must be isomorphic to the Rieffel-Wang deformation C(G)θ˜ in the sense of [31] of some compact group G which acts on M smoothly, where −θ 0 θ˜ = . 0 θ Proof: By passing to the reduced quantum group Qr if necessary, we can assume without loss of generality that the Haar state is faithful and then adapt the proof of Theorem 3.11 of [8] to see that Q−θ˜ has an action on C(M ) ∼ = (Mθ )−θ , which is smooth in the sense of the present paper. This implies Q−θ˜ is isomorphic with some C(G), hence Q ∼ = C(G)θ˜.✷

13

Some (counter)examples

In this section we study some examples to understand the necessity of the conditions of the Theorem 11.9. Example 1: Our first example is due to Huichi Huang ([19]) who showed that there exists a faithful action of a genuine compact quantum group on a compact, connected, metrizable topological space. Let us consider the example 3.10, (1) of his paper( [19]). Let Y = [0, 1], X4 = {1, 2, 3, 4} and Y1 = {0} in the notation of [19] and S = X4 × Y / ∼ obtained by taking wedge sum of 4 unit intervals identifying (xi , 0) for all i = 1, . . . , 4. Observe that the space fails to be smooth precisely at the point where those 4 copies are glued together topologically. The action 67

ˆ 4 of quantum permutation group for n = 4 (which is non α : C(S) → C(S)⊗A commutative) is given by: X f ⊗ ei ⊗ qij f orf ∈ C(Y ). α(f ) = ij

Pick an interior point (i, y) ∈ X4 × Y where i is between 1 and 4 and y ∈ (0, 1). d At this point we can apply the vector field dt in the direction of the ith copy of Y (this is just taking derivative in (0, 1)). (

X ′ d f (y)qij ⊗ id)(α(f ))(i, y) = dt j α(g)(i, y) =

X

g(y)qij .

j

d d Thus, ( dt ⊗ id)α(f )α(g) = α(g)( dt ⊗ id)α(f ), at all the points where the space is smooth (so that we can make sense of a smooth vector field in the neighbourhood of the points).This demonstrates the necessity of smoothness condition of theorem 11.9.

Example 2: Let us give an example of Hopf-∗ algebra corresponding to a genuine locally compact quantum group having a nice action on a noncompact manifold, namely R, in the algebraic setting. Let A ≡ C[x] be the * algebra of polynomials in one variable with complex coefficients and Q0 be the Hopf * algebra generated by a, a−1 , b subject to the following relations aa−1 = a−1 a = I, ab = q 2 ba, where q is a parameter as described in [46]. The coproduct is given by (see [46] for details) ∆(a) = a ⊗ a, ∆(b) = a ⊗ b + b ⊗ I. We have a coaction α : k[x] → k[x] ⊗ Q0 given by α(x) = x ⊗ a + 1 ⊗ b. The algebra A is the algebraic geometric analogue of C ∞ (R) and we have the d following canonical derivation δ : A → A corresponding to the vector field dt of R: ′ δ(p) = p , ′

where p denotes the usual derivative of the polynomial p. However an easy computation gives (δ ⊗ id)α(x) = 1 ⊗ a, which do not commute with α(x) as ab 6= ba. 68

So (δ ⊗ id)α(x)α(x) 6= α(x)(δ ⊗ id)α(x). This means we don’t have a lift of this action to the bimodule of one-forms as a representation, unlike the case of smooth action of CQG on compact manifolds. This suggests that one may have genuine locally compact quantum group acting smoothly on connected non-compact manifolds. Acknowledgement: One of the authors (Debashish Goswami) would like to thank Prof. Marc A. Rieffel for inviting him to visit the Department of Mathematics of the University of California at Berkeley, where some of the initial ideas and breakthroughs of the work came. All the authors also thank Jyotishman Bhowmick for pointing out the work of Etingof and Walton ([14]).

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