DEFORMATIONS OF INFINITE PROJECTIONS ETIENNE BLANCHARD Abstract. Let A = (Ax ) be a (semi-)continuous field of C∗ -algebras over a compact Hausdorff space X and let p = (px ) be a projection in A such that each px ∈ Ax is properly infinite (x ∈ X). We prove that p ⊕ . . . ⊕ p (l summands) is properly infinite in Ml (A) for large enough l ∈ N if the C(X)-algebra A is upper semi-continuous. But this does not hold anymore if A is lower semi-continuous.
1. Preliminaries A powerful tool in the classification of C∗ -algebras is the study of their projections. Two projections p, q in a C∗ -algebra A are said to be Murray-von Neumann equivalent (respectively (resp.) p dominates q) if there exists a partial isometry v ∈ A with v ∗ v = p and vv ∗ = q (resp. v ∗ v ≤ p and vv ∗ = q). For short we write p ∼ q (resp. q 4 p). The non-zero projection p is said to be infinite (resp. properly infinite) if p is equivalent to a proper subprojection q < p (resp. p is equivalent to two mutually orthogonal projections p1 , p2 with p1 + p2 ≤ p) and p is finite otherwise. J. Cuntz introduced the following generalization: A positive element a in A dominates another positive element b in A (written b - a) if and only if (iff) there exists a sequence {dn }n in A such that d∗n adn → b ([16]). Further a ∈ A+ is called infinite (resp. properly infinite) iff there exists a non-zero positive element b in A such that a ⊕ b - a ⊕ 0 in M2 (A) (resp. a ⊕ a - a ⊕ 0 in M2 (A)). And a is said to be finite if a is not infinite. Kirchberg and Rørdam proved that that these definitions coincide with the ones given in the previous paragraph in case a is a projection ([16, Lemma 3.1]). Now a C∗ -algebra A is said to be infinite (resp. properly infinite) iff all strictly positive elements in A are infinite (resp. properly infinite). It is said to be finite (resp. stably finite) if all strictly positive elements in A are finite (resp. all strictly positive elements in Mn (A) are finite for all positive integer n). In order to study deformations of such algebras, let us recall a few notions from the theory of C(X)-algebras. Let X be a compact Hausdorff space and C(X) the C∗ -algebra of continuous functions on X with values in the complex field C. Definition 1.1. A C(X)-algebra is a C∗ -algebra A endowed with a unital ∗–homomorphism from C(X) to the centre of the multiplier C∗ -algebra M(A) of A. For all x ∈ X, we denote by evx : C(X) → C the evaluation map f 7→ f (x), by Cx (X) the ideal of functions f ∈ C(X) satisfying f (x) = 0, by Ax the quotient of A by the closed ideal Cx (X)A and by ax the image of an element a ∈ A in the fibre Ax . 1
Then the function (1.1)
N (a) : x 7→ kax k = inf{k [1 − f + f (x)]ak ; f ∈ C(X)}
is upper semi-continuous by construction. The C(X)-algebra is said to be continuous (or to be a continuous C∗ -bundle over X) if the function x 7→ kax k is actually continuous for all element a in A. Definition 1.2. ([5]) Given a continuous C(X)-algebra B, a C(X)-representation of a C(X)-algebra A on B is a C(X)-linear map π from A to the multiplier C∗ -algebra M(B) of B. Further π is said to be a continuous field of faithful representations if, for all x ∈ X, the induced representation πx of the fibre Ax in M(Bx ) is faithful. Note that the existence of such a continuous field of faithful representations π implies that the C(X)-algebra A is continuous since the function (1.2)
x 7→ kπx (ax )k = sup{k(π(a)b)x k ; b ∈ B such that kbk ≤ 1}
is lower semi-continuous for all a ∈ A. Conversely, any separable continuous C(X)-algebra A admits a continuous field of faithful representations. More precisely, there always exists a unital positive C(X)linear map ϕ : A → C(X) such that all the induced states ϕx on the fibres Ax are faithful ([4]). By the Gel’fand-Naimark-Segal (GNS) construction this gives a continuous field of faithful representations of A on the continuous C∗ -bundle of compact operators K(E) on the Hilbert C(X)-module E = L2 (A, ϕ). A simple C∗ -algebra A is purely infinite iff every non-zero hereditary C∗ -subalgebra B ⊂ A contains an infinite projection ([9]). Possible generalisations to the non-simple case are the following: – A C∗ -algebra A is said to be purely infinite (p.i.) iff A has no non-zero character and for all a, b ∈ A+ , ε > 0, with b in the closed ideal of A generated by a, there exists an element d ∈ A with kb − d∗ adk < ε ([16]). – A C∗ -algebra A is said to be locally purely infinite (l.p.i.) iff for all b ∈ A and all ideal J / A with b 6∈ J, there exists a stable C∗ -subalgebra DJ ⊂ b∗ Ab such that DJ 6⊂ J. Note that a C∗ -algebra A is p.i. iff for all b ∈ A, there exists a stable C∗ -subalgebra D∼ = D ⊗ K contained in the hereditary C∗ -subalgebra b∗ Ab such that for all (closed two sided) ideal J / A with b 6∈ J, then D 6⊂ J ([21, prop. 5.4]). Hence, every p.i. C∗ -algebra is l.p.i. ([8, prop. 4.11]). We shall study in this article a few problems linked to the converse implication. The author is grateful to E. Kirchberg, M. Rørdam and M. Dadarlat for helpful comments. He would also like to thank the Humboldt University and the Purdue University for invitations during which part of that work was written. 2. Continuous fields of properly infinite C∗ -algebras In this section, we study the stability properties of proper infiniteness under (upper semi-)continuous deformations. For all integer n ≥ 1, Mn (C) is the C∗ -algebra linearly generated by n2 operators 2
{ei,j } satisfying the relations ei,j ek,l = δj,k ei,l and (ei,j )∗ = ej,i (1 ≤ i, j ≤ n). The Cuntz C∗ -algebra On (resp. Tn ) is the unital C∗ -algebra generated by n isometries s1 , . . . , sn satisfying the relation s1 s∗1 + . . . + sn s∗n = 1 (resp. s1 s∗1 + . . . + sn s∗n ≤ 1). Remarks 2.1. a) The universal C∗ -algebra Tn is called En by Cuntz in [9]. b) A unital C∗ -algebra A is properly infinite iff there exists a properly infinite projection p ∈ A which is full in A, i.e. the closed two sided ideal in A generated by p equals A. Indeed, if p is such a projections, there exist a contraction w ∈ A s.t. 1 = w∗ pw and partial isometries t1 , t2 which generate a unital copy of T2 in pAp. Then the two isometries t1 w and t2 w generate a unital copy of T2 in A. Proposition 2.2. Let X be a compact Hausdorff space and let D be a unital separable C(X)-algebra the fibres of which are properly infinite. Then Ml (D) is properly infinite for some integer l > 0. Let us first prove the following lemma which is essentially contained in [8]. Lemma 2.3. Let n ≥ 2 be an integer. Fix n + 1 isometries s1 , . . . , sn+1 generating the C∗ -algebra Tn+1 . Let Tn be the unital copy of Tn generated by s1 , . . . , sn and define the projection pn := s1 s∗1 + . . . + sn s∗n . a) Let B be a unital C∗ -algebra and σ1 , σ2 unital ∗-homomorphisms from Tn+1 to B. Then the projections σ1 (pn ) and σ2 (pn ) are unitarily equivalent in B. b) Let A, B be unital C∗ -algebras, π : A B a unital ∗-epimorphism, θ : Tn+1 → A and σ : Tn+1 → B unital ∗-homomorphisms. Then there is a ∗-homomorphism θ0 : Tn → M2 (A) such that (ı ⊗ π)θ0 (r) = e1,1 ⊗ σ(r) for all r ∈ Tn and θ0 (1) is full in M2 (A). c) Suppose that the C∗ -algebra A is the pullback of the two unital C∗ -algebras A1 and A2 along the ∗-epimorphisms πk : Ak B (k = 1, 2). If θk : Tn+1 → Ak are unital ∗-homomorphisms (k = 1, 2), then there exists a unital ∗-homomorphism θe = (θe1 , θe2 ) : Tn → M2 (A) ⊂ M2 (A1 ) ⊕ M2 (A2 ), i.e. M2 (A) is properly infinite. Proof. a) The projection pn is full in Tn+1 . Hence, the two full properly infinite projections σ1 (pn ) and σ2 (pn ) are Murray-von Neumann equivalent in B ([9, Theorem 1.4], [8, Lemma 4.15]). The projection 1 − pn ∈ Tn+1 is also full and properly infinite since 1 − pn ≥ sn+1 s∗n+1 ≥ (sn+1 s1 )(1 − pn )(sn+1 s1 )∗ + (sn+1 s2 )(1 − pn )(sn+1 s2 )∗ . Thus, there exists by [17, Proposition 2.2.2] a unitary v ∈ B satisfying v ∗ σ1 (pn ) v = σ2 (pn ) . b) One can find a unitary P v ∈ B satisfying v ∗ πθ(pn )v = σ(pn ) by a). Define now the unitary u = 1B − σ(pn ) + 1≤k≤n σ(sk )v ∗ πθ(s∗k ) v in U(B). Then σ(sk ) = u v ∗ πθ(sk ) v
for all k ∈ {1, . . . , n}.
Take unitary liftings u e and ve in U(M2 (A)) of the unitaries u⊕u∗ and v ⊕v ∗ which are in 0 ∗ the connected component of the identity. The formulae θ (sk ) = u e ve e1,1 ⊗θ(sk ) ve (1 ≤ 0 k ≤ n) define a ∗-homomorphism θ from Tn to M2 (A). Further, θ0 (1) = ve∗ e1,1 ⊗ 1)e v is full in M2 (A) 3
c) The pullback A is (isomorphic to) the C∗ -subalgebra of A1 ⊕ A2 of pairs (a1 , a2 ) satisfying π1 (a1 ) = π2 (a2 ). And π1 θ1 (pn ) = v ∗ π2 θ2 (pn )v for some v ∈ U(B) by a). If we set θ20 (r) := e1,1 ⊗ θ2 (r) ∈ M2 (A2 ) for all r ∈ Tn , there is a ∗-homomorphism θ10 : Tn → M2 (A) s.t. (ı ⊗ π1 )θ10 = (ı ⊗ π2 )θ20 by b), whence a ∗-homomorphism θ0 = (θ10 , θ20 ) from Tn to M2 (A). Besides, the properly infinite projection θ0 (1) is full in M2 (A) since the two projections θ10 (1) and θ20 (1) are full in M2 (A1 ) and M2 (A2 ). Thus, there exists an isometry z ∈ M2 (A) s.t. zz ∗ ≤ θ0 (1) by remark 2.1.b) and the map r 7→ z ∗ θ0 (r)z defines a unital ∗-homorphism from Tn to M2 (A). Proof of Proposition 2.2. For all x ∈ X there exist a open neighbourhood U (x) of x in X with closure U (x) and a unital ∗-homomorphism T2 → D|U (x) := D/C0 (X\U (x))D since T2 is semiprojective ([2, §4.7]) and the fibre Ax is properly infinite. Thus, there exist a finite covering X = U1 ∪ . . . ∪ Un of the compact space X by open subsets Ui and unital ∗-homomorphisms θi : T2 → D|Ui =: Ai (1 ≤ i ≤ n). Take a unital embedding of Tn in T2 . Then assertion c) of the above Lemma gives by induction a unital ∗-homomorphism θe : T2 → Ml (D) for l = 2n−1 . Remark 2.4. Uffe Haagerup indicated me another way to prove Proposition 2.2: If the unital C∗ -algebra D is stably finite C∗ -algebra, then there exists a bounded nonzero lower semi-continuous 2-quasi-trace on D ([13], [3, page 327]). Now, if D is also a C(X)-algebra for some compact Hausdorff space X, this implies that there is a bounded non-zero lower semi-continuous 2-quasitrace Dx → C for (at least) some point x ∈ X ([14, Prop. 3.7]). But then, the fibre Dx cannot be properly infinite. Question 2.5. Can one always assume l = 1 in Proposition 2.2? i.e. is a unital continuous C(X)-algebra D properly infinite if (and only if) all its fibres are properly infinite? In other words, are the two unitarily equivalent full properly infinite projections q1 = σ1 (pn ) and q2 = σ2 (pn ) defined in assertion a) of Lemma 2.3 always homotopic (in the set of all projections) in the C∗ -algebra B? Note that the answer to that question is negative if one only assumes the two full projections q1 and 1 − q1 to be infinite (see e.g. [2, Example 6.10.1]). Indeed, let T = { z ∈ C ; |z| = 1} and A = C(T3 ) ⊗ M2 (C). Then, there exists a unitary u ∈ U(A) s.t. the two projections q = 1 ⊗ e1,1 and uqu∗ are not homotopic in A (see e.g. [17, Example 11.3.4]). Hence, if B is the tensor product B = T1 ⊗ A, both the projections q1 = 1 ⊗ q and 1D − q1 = 1 ⊗ 1 ⊗ e2.2 are infinite in B. But the projections q1 and (1 ⊗ u)q1 (1 ⊗ u)∗ = 1 ⊗ uqu∗ cannot be homotopic in B because a composition with the quotient maps T1 → C(T) and ev1 : C(T) → C would imply that the projections q and uqu∗ are homotopic in A. 3. Lower semi-continuous fields of properly infinite C∗ -algebras Let us study whether the above results can be extended to lower semi-continuous (l.s.c.) C∗ -bundles (A, {σx }) over a compact Hausdorff space X. Recall that any such separable l.s.c. C∗ -bundle admits a faithful C(X)-linear representation on a Hilbert C(X)-module E such that, for all x ∈ X, the fibre σx (A) is isomorphic to the induced 4
image of A in L(Ex ) ([6]). Thus, the problem boils down to the following: Given a separable Hilbert C(X)-module E with infinite dimensional fibres Ex , the unit p of the C∗ -algebra LC(X) (E) of bounded adjointable C(X)-linear operators acting on E has a properly infinite image in L(Ex ) for all x ∈ X. But is the projection p itself properly infinite in LC(X) (E)? Dixmier and Douady have proved that this is always the case if the space X has finite topological dimension ([11]). But it does not hold anymore in the infinite dimensional case ([11, §16, Corollaire 1], [20]), even if X is contractible ([7, Corollary 3.7]). References [1] S. Armstrong, K. Dykema, R. Exel, H. Li, On embeddings of full amalgamated free product C ∗ algebras, Proc. Amer. Math. Soc. 132 (2004), 2019–2030. [2] B. Blackadar, K-theory for Operator Algebras, MSRI Publications 5, Cambridge Univ. Press (1998). [3] B. Blackadar, D. Handelman, Dimension functions and traces on C∗ -algebras, J. Funct. Anal. 45 (1982), 297–340. [4] E. Blanchard, D´eformations de C∗ -alg`ebres de Hopf. Bull. Soc. Math. France 24 (1996), 141–215. [5] E. Blanchard, Tensor products of C(X)-algebras over C(X), Ast´erisque 232 (1995), 81–92. [6] E. Blanchard, A few remarks on C(X)-algebras, Rev. Roumaine Math. Pures Appl. 45 (2001), 565–576. [7] E. Blanchard, E. Kirchberg, Global Glimm halving for C∗ -bundles, J. Op. Th. 52 (2004), 385–420. [8] E. Blanchard, E. Kirchberg, Non-simple purely infinite C∗ -algebras: the Hausdorff case, J. Funct. Anal. 207 (2004), 461–513. [9] J. Cuntz, K-theory for certain C∗ -algebras, Ann. of Math. 113 (1981), 181–197. [10] M. Dadarlat, Continuous fields of C∗ -algebras over finite dimensional spaces, preprint. [11] J. Dixmier, A. Douady, Champs continus d’espaces hilbertiens et de C∗ -alg`ebres, Bull. Soc. Math. France 91 (1963), 227–284. [12] K. Dykema, D. Shlyakhtenko, Exactness of Cuntz-Pimsner C∗ -algebras, Proc. Edinb. Math. Soc. 44 (2001), 425–444. [13] D. Handelman, Homomorphism of C∗ -algebras to finite AW ∗ algebras, Michigan Math J. 28 (1981), 229-240. [14] I. Hirshberg, M. Rørdam, W. Winter, C0 (X)-algebras, stability and strongly self-absorbing C*algebras, Preprint July 2006. [15] G.K. Pedersen, Pullback and pushout constructions in C∗ -algebra theory, J. Funct. Anal. 167 (1999), 243–344. [16] E. Kirchberg, M. Rørdam, Non-simple purely infinite C∗ -algebras, Amer. J. Math. 122 (2000), 637–666. [17] F. Larsen, N. J. Laustsen, M. Rørdam, An Introduction to K-theory for C∗ -algebras, London Mathematical Society Student Texts 49 (2000) CUP, Cambridge. [18] N. C. Phillips A classification theorem for nuclear purely infinite simple C ∗ -algebras., Doc. Math. 5 (2000), 49–114 [19] M. Rørdam, Classification of nuclear, simple C ∗ -algebras. Classification of nuclear C ∗ -algebras. Entropy in operator algebras, 1–145, Encyclopaedia Math. Sci., 126, Springer, Berlin, 2002. [20] M. Rørdam, A simple C ∗ -algebra with a finite and an infinite projection, Acta Math. 191 (2003), 109–142. [21] M. Rørdam, Stable C*-algebras, Advanced Studies in Pure Mathematics 38 “Operator Algebras and Applications” (2004), 177-200.
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