TYPE II NON COMMUTATIVE GEOMETRY. I. DIXMIER TRACE IN

arXiv:math/0012233v3 [math.KT] 12 Mar 2003

TYPE II NON COMMUTATIVE GEOMETRY. I. DIXMIER TRACE IN VON NEUMANN ALGEBRAS MOULAY-TAHAR BENAMEUR AND THIERRY FACK Abstract. We define the notion of Connes-von Neumann spectral triple and consider the associated index problem. We compute the analytic Chern-Connes character of such a generalized spectral triple and prove the corresponding local formula for its Hochschild class. This formula involves the Dixmier trace for II∞ von Neumann algebras. In the case of foliations, we identify this Dixmier trace with the corresponding measured Wodzicki residue.

Contents Introduction 1. Dixmier traces on von Neumann algebras 1.1. Review of the classical case 1.2. Dixmier ideal in a von Neumann algebra 1.3. Dixmier trace 1.4. Dixmier trace and residue of zeta functions 2. The von-Neumann index problem 2.1. Von Neumann spectral triples 2.2. Regularization formulae 2.3. Index theory in von Neumann algebras 2.4. The index map associated with a spectral triple 3. Measured foliations 4. The local positive Hochschild class Appendix A. Singular numbers A.1. τ -measurable operators A.2. τ -singular numbers A.3. Non commutative integration theory References

1 4 4 4 5 8 11 11 13 15 17 22 25 31 31 31 32 33

Introduction This paper is devoted to an extension, in the framework of type II von Neumann algebras, of the notion of spectral triple introduced by A. Connes [15] in noncommutative geometry. Recall that a spectral triple is a triple (A, H, D) where H is a Hilbert space, A is a ∗−subalgebra of B(H) and D = D∗ is an unbounded operator on H whose resolvent is compact and which interracts with A in a suitable way. A. Connes showed that a large part of Riemannian geometry may be recovered from the study of specific spectral triples. More precisely, let M be a compact oriented (spin) Riemannian n−manifold and denote by: • H the Hilbert space of L2 −spinors; • A the ∗−algebra of smooth functions on M ; • D is the L2 −extension of the Dirac operator on M . 1

2

M.-T. BENAMEUR AND T. FACK

From (A, H, D), we can recover (1) The Riemannian metric d on M , by the formula: d(x, y) = Sup{|f (x) − f (y)|, f ∈ A and k[D, f ]k ≤ 1}; (2) The smooth structure of A, since we have for any continuous function f on M : \ f ∈ A ⇐⇒ f ∈ Dom(δ n ), n≥1

where δ is the unbounded derivation on B(H) defined by δ(T ) := [|D|, T ]; (3) The fundamental cycle of M , by the formula: Z f 0 df 1 · · · df n = Trω (f 0 [D, f 1 ] · · · [D, f n ](1 + D2 )−n/2 ), M

where Trω is the Dixmier trace [23, 15]. In general, spectral triples (A, H, D) give rise to morphisms from the K−theory group K∗ (A) to the integers. Spectral triples are called even triples when the Hilbert space is Z2 −graded with A even and D odd for the grading. In this case, the corresponding map on K−theory is defined on K0 (A) and assigns to any idempotent e = e2 ∈ MN (A), the Fredholm index Ind([e(F ⊗ 1N )e]+ ) of the positive part of e(F ⊗ 1N )e, where F is the sign of D. This map K0 (A) → Z is given by: Ind([e(F ⊗ 1N )e]+ ) = φ2k (e, · · · , e). where φ2k (k large enough), is the cyclic 2k−cocycle on A defined by φ2k (a0 , · · · , a2k ) := (−1)k Tr(γa0 [F, a1 ] · · · [F, a2k ]), where γ denotes the grading involution on H. See [17] for more details. Denote by Ch(A, H, D) the Chern-Connes character of the spectral triple (A, H, D), i.e. the image of φ2k (in the even case) in the periodic cyclic cohomology of A. The computation of Ch(A, H, D) in terms of local data involving appropriate noncommutative residues is the main step toward the solution of the index problem associated with (A, H, D), and was carried out by A. Connes and H. Moscovici in [19, 20]. In this paper, we consider an extended notion of spectral triples where the operator D is affiliated with some semi-finite von Neumann algebra M. The unitary group of the commutant of M is thus a symmetry group for D that we want to take into account in order to discuss the associated index problem. The resolvent of D should then be compact with respect to the trace τ of M, whose dimension range can be [0, +∞]. Such triples are called Connes-von Neumann spectral triples here. The Murray-von Neumann dimension theory [22] allows to associate a natural index problem to any Connes-von Neumann spectral triple (A, M, D), and our goal is to extend the Connes-Moscovici local index theorem to this framework. This will give a Non Commutative Geometry approach to many well-known index problems involving von Neumann algebras, such as measured foliations [13], Galois coverings [1] or almost periodic operators [12, 40]. Our approach is also motivated by some applications to statistical mechanics that we have in mind [29]. The present paper is a first of a series where we prove the local index theorem for von-Neumann spectral triples. In order to handle the locality in our discussion of the index problem, we were naturally led to introduce the notion of Dixmier trace for operators affiliated with a semi-finite von Neumann algebra. In particular, we show that the relevent ideal of infinitesimals is Z t L1,∞ (M, τ ) := {T ∈ M/ µs (T )ds = O(Log(t)) when t → +∞}, 0

where µs (T ) (s > 0) are the generalized s−numbers of T [25]. The Dixmier trace of a positive element T ∈ L1,∞ (M, τ ) is then defined by: Z t 1 µs (T )ds , τω (T ) := lim t→ω Log(1 + t) 0

VON NEUMANN DIXMIER TRACE

3

where limt→ω f (t) is an appropriate conformal invariant limiting process. For a foliation (M, F ) with an invariant transverse measure Λ, we recover the transverse integration from the Dirac operator along the leaves D by using our Dixmier trace: Z f dΛν . τωΛ (f |D|−p ) = C(p) M

In this formula, which may be viewed as the natural extension of Connes’ formula [15], Λν is the measure on M associated with Λ [13], and C(p) is a constant depending only on the leaf dimension p. Let us now describe more precisely the contents of this paper. In the first section, we introduce the Dixmier trace for semi-finite von Neumann algebras and we indicate its relationship with residues of zeta functions. In Section 2, we define the notion of (p, ∞)−summable von Neumann spectral triple. We show that such a von Neumann spectral triple (A, M, D) gives rise to a noncommutative integral Z M ∋ T 7−→ ω T = τω (T |D|−p ) ∈ C, which is a hypertrace on the algebra generated by A and [D, A]. We also express this noncommutative integral by various regularized spectral formulae such as Z 1 1 −(D/λ1/p )2 τ (T e ) , lim ωT = Γ( p2 + 1) λ→ω λ which extends the famous Weyl formula. Then, by using the Murray-von Neumann dimension theory, we define the index map associated with (A, M, D). This index map is described by an analytical cyclic cocycle on A, the Chern-Connes character of (A, M, D), that we identify by proving a generalized Calderon type formula. Section 3 is devoted to a careful analysis of the natural von-Neumann spectral triples associated with order one differential operators along the leaves of a measured smooth foliation on a compact manifold. For an order −p pseudodifferential operator along the leaves P = (PL ), we then compute the Dixmier trace τω (P ) and show that it coincides with the integrated leafwise Wodzicki residue resL (PL ). We also relate the computation of the analytical Chern-Connes character constructed in Section 2, to the solution of the measured index theorem for foliations [13, 34]. In Section 4, we prove a local formula for the image of the Chern-Connes character in Hochschild cohomology. More precisely, we prove that the pairing of this image with Hochschild cycles on A is the same as the pairing of the Hochschild cocycle φ given by the following local formula in the even case: XZ X i i i ai0 ⊗ · · · ⊗ aip >= < φ, ω γa0 [D, a1 ] · · · [D, ap ]. i

i

In the case of measured foliations, the Hochschild class of the Chern-Connes character of the Dirac operator along the leaves coincides with the Ruelle-Sullivan current. When the leaves are four dimensional, our local formula also furnishes, as in [15], a lower bound for the measured Yang-Mills action YMΛ (∇E ) associated with any compatible connection ∇E on a hermitian vector bundle E over the foliated manifold:

c1 (E)2 − c2 (E), [CΛ ] > | ≤ YMΛ (∇E ), 2 where CΛ is the Ruelle-Sullivan current associated with the transverse Λ [38]. For the convenience of the reader, we have also added an appendix on the von Neumann singular numbers. |
0 {kT Ek, E = E ∗ = E 2 ∈ M, τ (1 − E) ≤ t} the tth generalized s-number of T (See Appendix A.1 for more details). An element T ∈ M is called τ -compact if limt→∞ µt (T ) = 0. The set of all τ -compact elements in M is a norm closed ideal of M that we shall denote by K(M, τ ). By [25][page 304], the ideal K(M, τ ) is the norm closure of the ideal R(M, τ ) of all elements X in M whose final support r(X) = Supp(X ∗ ) satisfies τ (r(X)) < ∞.


5

Definition 1. An element T ∈ M is called of Dixmier trace class (with respect to τ ) if: Z t µs (T )ds = O(Log(1 + t)), when t → +∞. 0

Rt In the sequel, we shall set σt (T ) := 0 µs (T )ds. The set of Dixmier trace class operators is a vector space that we shall denote by L1,∞ (M, τ ). It is a Banach space for the norm kT k1,∞ = Supt>0

σt (T ) , Log(t + 1)

and an ideal in M which contains L1 (M, τ ) ∩ M. Note that we have for any T ∈ L1,∞ (M, τ ) and t > 0: µt (T ) ≤

Log(1 + t) σt (T ) ≤ kT k1,∞ , t t

so that we get for any ǫ > 0: L1 (M, τ ) ∩ M ⊂ L1,∞ (M, τ ) ⊂ L1+ǫ (M, τ ) ∩ M ⊂ K(M, τ ). We shall set for 1 < p < +∞: 1

Lp,∞ (M, τ ) := {T ∈ M/σt (T ) = O(t1− p )}. For T ∈ Lp,∞ (M, τ ) with 1 < p < +∞, we have µt (T ) = O(t−1/p ), it follows that |T |p ∈ L1,∞ (M, τ ). 1.3. Dixmier trace. The general notion of singular traces on von Neumann algebras has been introduced in [27] and used in [28] to investigate the Novikov-Shubin invariants. For our purpose, we shall consider here Dixmier traces defined in terms of singular numbers. To this end, we shall use limiting processes ω : L∞ ([0, +∞[) ∋ f 7−→ ω(f ) ∈ C. More precisely, ω is a linear form on L∞ ([0, +∞[)) satisfying the following conditions: (1) lim ess inf t→+∞ f (t) ≤ ω(f ) ≤ lim ess sup t→+∞ f (t); Rt 1 (2) ω(f ) = ω(M (f )), where M (f )(t) = Log(t) f (s)ds/s. 1

Note that M (f ) is continuous and bounded on [1, +∞[ for any bounded function f . The first condition implies that ω is a state on L∞ ([0, +∞[)) that vanishes on C0 ([0, +∞[), and the second condition implies the following scale-invariance property: (3) For any λ > 0 and any f ∈ L∞ ([0, +∞[), we have ω(f ) = ω(fλ ), where fλ (t) = f (λt). The existence of a limiting process satisfying the two conditions is obvious. Indeed, let φ be a state on the C ∗ -algebra Cb (R+ ) vanishing on C0 (R+ ) and set for a = (an )n≥0 ∈ l∞ (N): lim(a) = φ(˜ a), φ

where a ˜ ∈ Cb ([0, +∞[) is the piecewise linear function defined by a ˜(n) =

a0 +···+an n+1

for any n ≥ 0. Then,

ω(f ) := lim(φ(M n (f )),

(1)

φ

is a limiting process satisfying (1) and (2). As in [15], we shall write ω(f ) = limt→ω f (t). Note that we have, for such a limiting process: (4) | lim f (t) − α lim f (λtα )| ≤ (1 − α)kf k∞ , t→ω

t→ω

∞

for any α ∈]0, 1[, any λ > 0 and any f ∈ L ([0, +∞)). This follows from the estimate: Z t Z t α | Log(λ)| 1 . f (s)ds/s − f (λsα )ds/s| ≤ kf k∞ (1 − α + 2) | Log(t) 1 Log(t) 1 Log(t)

6


Definition 2. For any limiting process ω as above, the Dixmier trace τω (T ) of a positive operator T ∈ L1,∞ (M, τ ) is defined by: σt (T ) . τω (T ) := ω t → Log(t + 1) R t σs (T ) ds 1 Let us point out that this definition depends on the choice of ω. However, if l = limt→+∞ [ Log(t) ] a log(s) s exists, then τω (T ) = l. From now on we fix a limiting process ω(f ) = limt→ω f (t). As in the classical case (cf [19]) one may prove that we have for any positive operators T, S ∈ L1,∞ (M, τ ): τω (T + S) = τω (T ) + τω (S). This enables to extend τω to a positive linear form on the Dixmier ideal L1,∞ (M, τ ). A classical argument (cf [15]) shows that τω (ST ) = τω (T S) for any T ∈ L1,∞ (M, τ ) and any S ∈ M. Note that τω (T ) = 0 when T ∈ M ∩ L1 (M, τ ). In the same way, it is easy to check for instance by using the inequality Z Z t

t

µs (T )µs (S)ds,

µs (T S)ds ≤

0

0

(See [25]), that τω (T S) = 0 for any T, S ∈ L1,∞ (M, τ ).

Theorem 1. Let A ∈ L1,∞ (M, τ ) be a positive element and set Et := 1]t,+∞) (A) for any t > 0. Then for any T ∈ M, the functions 1 1 t 7→ τ (T Eµt (A) A) and t 7→ τ (T E1/t A), Log(t + 1) Log(t + 1) are bounded and we have: 1 τ (T Eµt (A) A) τω (T A) = lim t→ω Log(t + 1) 1 τ (T E1/t A) = lim t→ω Log(t + 1) Proof. Since µs (A) → 0 when s → +∞, we have: Z |τ (T Eµt (A) A)| ≤ kT k

µs (A)ds ≤ kT k

{s>0/µs (A)>µt (A)}

Z

t

µs (A)ds.

0

1 1 It follows that t 7→ Log(t+1) τ (T Eµt (A) A) is bounded. To prove that Log(t+1) τ (T E1/t A) is bounded, we may 1 assume w.l.o.g. that A 6= 0. For any t > kAk , let s(t) be the unique s ≥ 0 such that:

µs (A) ≤ 1/t and µs−ǫ (A) > 1/t, ∀ǫ > 0. Since A ∈ L (M, τ ), there exists a constant C > 0 such that µs (A) ≤ C Log(s+1) for any s > 0. Set s+1 u(t) = s(t) + 1. From the inequality 1,∞

1/t < µs(t)−ǫ (A) ≤ C

Log(s(t) − ǫ + 1) , s(t) − ǫ + 1

we deduce by letting ǫ → 0: u(t) ≤ Ct Log(u(t)) for t > 1/kAk. We claim that this implies the existence of a constant K > 0 such that: (2)

u(t) ≤ Kt Log(t + 1)

for t > 1/kAk.

Since Equation (2) is obvious when u is bounded, we may assume again w.l.o.g. that it is not the case, and hence limt→+∞ u(t) = +∞ since u is non-decreasing. Assume that (2) is false. Then there exists for any integer n > 0 a real number tn > 1/kAk such that: 1 n Log( + 1). u(tn ) > ntn Log(tn + 1) > kAk kAk


7

It follows that u(tn ) → +∞ when n → +∞ and hence tn → +∞. Write u(tn ) = ǫn tn Log(tn + 1) where ǫn > n. We have: C≥

ǫn Log(tn + 1) u(tn ) = −→ +∞, tn Log(u(tn )) Log(tn ) + Log(Log(tn + 1)) + Log(ǫn )

a contradiction. So, equation (2) is true and we have for t > 1/kAk: |τ (T E1/t A)|

≤

kT kτ (E1/t A) Z s(t) kT k µs (A)ds

=

0

≤

kT k

Z

Kt Log(t+1)

µs (A)ds

0

= a fact which implies that t 7→ Let us set, for any T ∈ M:

O(Log(t Log(t + 1)),

1 Log(t+1) τ (T E1/t A)

ϕ(T ) := τω (T A), ψ(T ) := lim

t→ω

is bounded.

τ (T E1/t A) τ (T Eµt (A) A) and θ(T ) := lim . t→ω Log(1 + t) Log(1 + t)

We thus define positive linear forms on M and we claim that (3)

θ ≤ ψ ≤ ϕ.

, there exists for For simplicity, set Pt = Eµt (A) and Qt = E1/t for t > 0. Since we have µt (A) ≤ C Log(1+t) 1+t any α ∈]0, 1[ a constant Cα > 0 such that µt (A) ≤ tα−1 /Cα for any t > 0. Hence: QCα t1−α A ≤ Pt A. For any T ∈ M with T ≥ 0, we deduce that: τ (T QCα t1−α A) ≤ τ (T Pt A), and hence, since

Log(t+1) Log(Cα t1−α +1)

→

1 1−α

when t → +∞:

(1 − α) lim f (Cα t1−α ) ≤ lim t→ω

where f (t) =

τ (T Qt A) Log(t+1) .

t→ω

τ (T Pt A) = ψ(T ), Log(t + 1)

But we have: | lim f (t) − (1 − α) lim f (Cα t1−α )| ≤ αkf k∞ , t→ω

t→ω

so that we get by letting α → 0: θ(T ) = lim f (t) ≤ ψ(T ). t→ω

On the other hand, since we have τ (Pt ) ≤ t (cf [26][Prop. 2.2, p. 274]), we get: Z +∞ Z t 1/2 1/2 1/2 1/2 τ (T Pt A) = τ (A T A Pt ) = µs (A T A Pt )ds = µs (A1/2 T A1/2 Pt )ds. 0

0

The last equality is deduced from the fact that [26][Lemma 2.6, p. 277]: µs (A1/2 T A1/2 Pt ) = 0, We thus have: τ (T Pt A) ≤ kPt k

Z

∀s ≥ τ (Pt ),

t

µs (A1/2 T A1/2 )ds,

0

and hence:

ψ(T ) ≤ τω (A1/2 T A1/2 ) = τω (T A) = ϕ(T ).

8


This proves (3). To show that θ = ψ = ϕ and achieve the proof of the theorem, it suffices to prove that ϕ(I) ≤ θ(I). But we have by [26][p. 289]: σt (A) = Inf{kT1 k1 + tkT2 k/A = T1 + T2 }, so that we get by taking T1 = Qt A and T2 = A − T1 : σt (A) ≤ τ (Qt A) + tk1[0,1/t] (A)k ≤ τ (Qt A) + 1. It follows that: ϕ(I) = lim

t→ω

τ (Qt A) σt (A) ≤ lim = θ(I). Log(t + 1) t→ω Log(t + 1)

Remark 1. In the above proof, the property ω(f ) = ω(M (f )) was used to prove that θ ≤ ψ. It is worthpointing out that in most of the examples we have in mind, the operator A will actually satisfy the following better estimate: µt (A) = O(1/t). In this case, the above theorem remains true for limiting processes ω(f ) = limω f (t) only satisfying the weaker scale invariance property ω(fλ ) = ω(f ). Indeed, if t > 0 and if A satisfies the relation µt (A) = C/t, we have Qt/C A ≤ Pt A, and hence τ (T Qt/C A) ≤ τ (T Pt A). So if we only assume that limt→ω f (t) = limt→ω f (t/C), we get θ(T ) = lim

t→ω

τ (T Qt/C A) τ (T Qt A) = lim ≤ lim τ (T Pt A) = ψ(T ). t→ω Log(t/C) t→ω Log(t)

1.4. Dixmier trace and residue of zeta functions. Let us first define the zeta function of a positive self-adjoint τ -discrete operator T in an infinite semi-finite von Neumann algebra A acting on H and equipped with a normal faithful positive trace τ . Definition 3. A positive self-adjoint τ -measurable operator T on H is called τ -discrete if (T − λ)−1 ∈ K(M, τ ) for any λ < 0. R +∞ It may be proved (cf [37][page 48]) that T = 0 λdEλ is τ -discrete if and only if one of the two following properties holds: (i) ∀λ ∈ R, τ (Eλ ) < +∞; (ii) ∃λ0 < 0 such that (T − λ0 )−1 ∈ K(M, τ ). For such a positive τ -discrete operator T , the function NT (λ) := τ (Eλ ) is well defined on R. Moreover, it is nondecreasing, positive and right continuous. R +∞ Definition 4. Let T = ǫ λdEλ be a positive self-adjoint τ -discrete operator with spectrum in [ǫ, +∞), where ǫ > 0. The zeta function ζT of T is defined by: Z +∞ ζT (z) := λz dNT (λ), ǫ

for any complex argument z such that the above integral converges.


9

Since we have for any R > 1: Z

R

λz dNT (λ) =

1

Z

Log(R)

ezt dα(t),

0

where α(t) = NT (et ), we know from the classical Laplace-Stieltjes transform theory (see [43]) that the integral Z +∞ λz dNT (λ) ǫ

converges for Re(z) < −dT and diverges for Re(z) > −dT , where: dT := limt→+∞

Log(α(t)) Log(NT (λ)) = limλ→+∞ . t Log(λ)

Moreover, ζT is analytic in the half-plane {Re(z) < −dT } and z = −dT is a singularity of ζT if dT < +∞. We also have: dT = Inf{µ ∈ R, T z ∈ L1 (M, τ ) for Re(z) < −µ}. Indeed, it follows from the normality of the trace that: Z R SupR>0 λx dNT (λ) = SupR>0 τ (ER (T )T x) = τ (T x ),

for any x ∈ R.

0

In particular: dT < +∞ ⇐⇒ ∃x ≤ 0 such that T x ∈ L1 (M, τ ). Sometimes, dT is called the quantum τ -dimension of the operator T . Theorem 2. Let T be a positive τ -discrete operator with spectrum in [ǫ, +∞], ǫ > 0. If 0 < dT < +∞, the following conditions are equivalent: (i) (x + dT )ζT (x) → A when x → −dT , x ∈ (−∞, −dT [; (ii) T −dT ∈ L1,∞ (M, τ ) and Z t 1 A (4) τω (T −dT ) = lim µs (T −dT )ds = − . t→+∞ Log(1 + t) 0 dT Proof. This theorem R ∞ easily follows from the equivalence, for any positive and non increasing function f on [0, +∞[ such that 0 f (t)s dt < +∞ for any s > 1, of the two following assertions: R +∞ (a) (s − 1) 0 f (t)s dt → L when s → 1+ ; R u 1 (b) Log(u) 0 f (s)ds → L when u → +∞. The proof of this equivalence uses classical abelian and tauberian theorems. For completeness, we give a proof based on Proposition 1 (see also [37]). (i) ⇒ (ii): Assume that Re(z) < −dT , and make the change of variable λ = eu/dT in the integral defining ζT (z). We get ζT (z) = f1 (−z/dT ) + f2 (z) where Z e Z +∞ −zu u/dT )1(dT ,+∞) (u) and f2 (z) = λz dNT (λ). f1 (z) = e dφ(u), with φ(u) = NT (e ǫ

0

The function f2 is entire, while f1 only converges for Re(z) > 1 and satisfies lim (x − 1)f1 (x) = −A/dT .

x→1+

Setting g(z) = f1 (z + 1) we then obtain Z ∞ Z t −zt g(z) = e dβ(t) where β(t) = e−u dφ(u) for any t ≥ 0. 0

0

10


Then g(z) is a convergent integral for Re(z) > 0 such that limx→0+ xg(x) = −A/dT . By the HardyLittlewood tauberian theorem, we get: Z t/dT β(t) 1 e A = lim = lim λ−dT dNT (λ), − t→+∞ t t→+∞ t e dT and hence: Z A 1 1 − = lim τ (1(1/t,∞) (T −dT )T −dT ). λdNT (λ−1/dT ) = lim t→+∞ t→+∞ dT Log(t) (1/t,+∞) Log(t) The result follows from Proposition 1. (i) ⇐ (ii): Let us first assume that A > 0. Fix ǫ > 0 with ǫ < dAT and choose M > 0 such that: Z t 1 | µs (T −dT )ds + A/dT | ≤ ǫ, ∀t ≥ M. Log(1 + t) 0 We have for any t ≥ M , Z t Z t Z t A A ds ds −dT (5) (− )ds ≤ (− − ǫ) µs (T + ǫ) ≤ . dT 1 + s d 1 +s T 0 0 0 If these inequalities were true for any t > 0 we would have: Z t Z t Z t f (s)ds ≤ h(s)ds ≤ g(s)ds, ∀t > 0, 0

0

0

where

1 1 −A −A − ǫ) + ǫ) , g(t) := ( and h(t) := µt (T −dT ) dT 1+t dT 1+t are non increasing positive functions. Using Polya’s inequality we would get: Z t Z t Z t a a ∀t > 0, ∀a ≥ 1, (f (s)) ds ≤ (h(s)) ds ≤ (g(s))a ds, f (t) := (

0

0

0

and hence by letting t → ∞:

Z +∞ (−A/dT − ǫ)a (−A/dT + ǫ)a ≤ µs (T −dT a )ds ≤ for any a > 1. a−1 a−1 0 Setting a = −x/dT with x < −dT and multiplying the above inequalities by −x − dT > 0, we get the result when x → −dT . Now since the inequaliy (5) is only true for t ≥ M , we take f1 , g1 and h1 equal to f, g and h for t ≥ M , and for t < M : Z M Z M Z M 1 1 1 f (v)dv, g1 (t) := g(v)dv and h1 (t) := h(v)dv. f1 (t) := M 0 M 0 M 0 Since f1 , g1 and h1 are nonincreasing positive functions such that: Z t Z t Z t g1 (s)ds, h1 (s)ds ≤ f1 (s)ds ≤ ∀t > 0, 0

0

0

we now use Polya’s inequality to get the conclusion. Indeed, the additional constants that arise with these modifications do not change the final computation of the residue of ζT at −dT . If A = 0 the same proof still works, replacing f by the zero function.

To end this paragraph, we point out the strong relation between the Dixmier trace and the asymptotics of the spectrum. For instance we have: Proposition 1. [37] Let T be as in Theorem 2. Assume that there exists δ > 0 such that ζT admits a meromorphic extension to {Re(z) < −dT + δ} with a simple pole at z = −dT . Then we have: Res−dT (ζT ) NT (λ) = τω (T −dT ). =− d T λ→+∞ λ dT lim


11

This proposition is a simple consequence of the Ikehara tauberian theorem. 2. The von-Neumann index problem The data proposed by A. Connes to define a ”geometry” is a triple (A, H, D), where A is a ∗-algebra represented in a Hilbert space H and D is an unbounded densely defined self-adjoint operator with a summability condition. To work with such a spectral triple, A. Connes introduces some constraints on the interaction between D and A. This formalism has been very fruitful especially in exploring index theory for singular spaces. We extend in this section some known results in non commutative geometry to the setting of von-Neumann algebras, whixh will allow us to reach the index theory of measured families of geometries. 2.1. Von Neumann spectral triples. In view of polynomial formulae, we shall restrict ourselves to finite dimensional spectral triples. The general case can be treated similarly extending the notion of θ-summability [6]. Definition 5. By a p−summable von Neumann spectral triple we shall mean a triple (A, M, D) where M ⊂ B(H) is a von Neumann algebra faithfully represented in a Hilbert space H and endowed with a (positive) normal semi-finite faithful trace τ , A is a ∗-subalgebra of the von Neumann algebra M, and D is a τ -measurable self-adjoint operator such that: (i) ∀a ∈ A, the operator a(D + i)−1 belongs to the Dixmier ideal Lp,∞ (M, τ ); (ii) Every element a ∈ A preserves the domain of D and the commutator [D, a] belongs to M; (iii) For any a ∈ A, the operators a and [D, a] belong to ∩n∈N Dom(δ n ), where δ is the unbounded derivation of M given by δ(b) = [|D|, b]. When M is Z2 -graded with A even and D odd, we say that the von Neumann-spectral triple is even and denote by γ ∈ M the grading involution. Otherwise, the triple is called an odd triple. Examples. (1) Let M be a compact Riemannian manifold of dimension n. Let D be a generalized Dirac operator. Then we set A = C ∞ (M ), M = B(H), D) where H is the L2 -space of corresponding generalized spinors. With the operator D we get an n-summable von Neumann spectral triple. It is even when n is even. As proved by A. Connes, one completely recovers the Gauss-Riemann calculus on M from the study of the spectral triple (A, M, D) [15]. (2) Let (M, F ) be a compact foliated manifold with a holonomy invariant transverse measure Λ and let p ≥ 1 be the dimension of the leaves of (V, F ). Let D be a generalized Dirac operator along the leaves of (M, F ) acting on sections of a hermitian vector bundle E. Denote by WΛ∗ (M, F ; E) the von Neumann algebra associated with Λ and E (see Section 3). The holonomy invariant transverse measure Λ gives rise to a semi-finite normal trace τΛ on WΛ∗ (M, F ; E) by the formula: Z τΛ (T ) := Trace(TL )dΛ(L). M/F

∞

(M ), WΛ∗ (M, F ; E), D)

The triple (A = C is then a p−summable von Neumann spectral triple which is not a type I spectral triple (See Section 3 for more details). Again the triple is even when p is even. ˜ → M be a Galois cover of a compact n−dimensional manifold M . Let D be the Γ (3) Let Γ ֒→ M cover of a generalized Dirac operator on M . Consider the von Neumann algebra M of bounded Γ-invariant operators defined by Atiyah in [1], with its natural trace TrΓ . Then (A = C ∞ (M ), M, D) is an n-summable von Neumann spectral triple (See [6] for more details), which is even when n is even. P (4) Let D = i ai (x)Dxi + b(x) be a first order uniformly elliptic differential operator with almost periodic ˜ coefficients on Rn . The index of such an operator can be defined by considering a spectral triple (A, M, D) ˜ is a direct integral over the Bohr compactification that we shall briefly describe, see [39]. The operator D ˜ x defined by: RnB of operators D X ˜x = D ai (x + y)Dyi + b(x + y).

˜ acts on L2 (Rn × Rn ). The algebra A is the algebra CAP∞ (Rn ) of smooth almost periodic functions So D B on Rn , and M is the von Neumann crossed product algebra L∞ (Rn ) ⋊ Rndiscrete which is a II∞ factor as

12


proved in [39]. Let (A, M, D) be a p−summable von-Neumann spectral triple. When D is invertible, we define the non commutative integral of T ∈ M by the formula: Z −p ). ω T := τω (T |D|

When D is not invertible, we replace for instance |D| by (D2 + 1)1/2 . Note that |D| − (D2 + 1)1/2 is bounded. For simplicity, we shall usually assume that D is invertible, see also the next section. We point out the following useful proposition. R Proposition 2. The map T 7→ ω T is a hypertrace on the algebra A˜ generated by A and [D, A] = {[D, a], a ∈ A}, i.e. Z Z ˜ ∀T ∈ M and A ∈ A. ω AT = ω T A,

T Proof. The algebra A˜ lies in n≥0 Dom(δ n ), where δ is as before the unbounded derivation T 7→ [|D|, T ]. ˜ We have for any A ∈ A: Z Z −p ) AT − ω T A = τω ([A, T ]|D| ω −τω (T [|D|−p , A]). Pp−1 −k Assume first that p is an integer. Then [|D|−p , A] = [|D|−1 , A]|D|−p+k+1 and [|D|−1 , A] = k=0 |D| −1 −1 −|D| [|D|, A]|D| . Therefore we get: =

−T [|D|−p , A] =

p−1 X

T |D|−k−1 [|D|, A]|D|−p+k .

k=0

On the other hand, for any k ∈ {0, · · · , p − 1}, the operator T |D|−k−1 [|D|, A]|D|−p+k is trace-class since T and [|D|, A] are bounded operators, and |D|−p−1 is trace class. Therefore τω (T [|D|−p , A]) = 0. The proof is thus complete when p is an integer. Now if p 6∈ N, we choose an integer k and a real number r ∈]0, 1[ such that p = rk. Then an easy computation shows that we have: [|D|−p , A] = −

k X

|D|−rm [|D|r , A]|D|−r(k−m+1) .

m=1 −α

Therefore, it suffices to show that τω (||D| S|D|−β |) = 0, where α = rm, β = r(k −m+1) and S = [|D|r , A]. But the operator S is bounded. Indeed, one can for instance use the integral expression of |D|r given by: Z +∞ r |D|(tI + |D|)−1 tr−1 dt, |D| = C 0

to deduce that for t ≤ 1, the operator |D|(tI + |D|)−1 is bounded with norm ≤ 1, while for t ≥ 1, one can use the relation [|D|(tI + |D|)−1 , A] = [|D|, A](tI + |D|)−1 − |D|(tI + |D|)−1 [|D|, A](tI + |D|)−1 , to conclude that S is bounded as allowed. Now the end of the proof goes as follows. By [26][Theorem 4.2, p. 286], we have: Z t Z t µs (|D|−α S|D|−β )ds ≤ kSk µs (|D|−(α+β) )ds, 0

0

and hence:

τω (||D|−α S|D|−β |) ≤ kSkτω (|D|−(p+r) ) = 0. The last equality is a consequence of the summability of the operator |D|−(p+r) .


13

We point out that for a given p−summable von-Neumann spectral triple (A, M, D) with p > 1 and D invertible, we have by Theorem 1: Z 1 τ (T 1]µt (|D|−1 ),+∞[ (|D|−1 )|D|−p ) = lim ωT t→ω Log(t + 1) 1 = lim τ (T 1]t−1/p ,+∞[ (|D|−1 )|D|−p ), t→ω Log(t + 1) for any T ∈ M. R 2.2. Regularization formulae. Let us give regularization formulae for the non commutative integral ω T . To this end, we need the following lemma. Lemma 1. Let (A, M, D) be a (p, ∞)−summable von Neumann spectral triple with p > 1 and D invertible. R +∞ Let f : [0, +∞) → C be a C 1 function such that limt→+∞ f (t) = 0 and 1 Log(t)|f ′ (t)|dt < +∞. Then, we have for any T ∈ M: Z |D| 1 τ (T f ( 1/p )|D|−p ). f (0) ω T = lim λ→ω Log(λ + 1) λ Proof. Let us first prove that the function

1 |D| τ (T f ( 1/p )|D|−p ), Log(λ + 1) λ is well defined and bounded. Since we have for any x ≥ 0, Z +∞ |f (x)| ≤ 1[0,t[ (x)|f ′ (t)|dt, λ 7−→

0

we deduce that: (6)

|f (

|D| )|D|−p )| ≤ λ1/p

Z

+∞

0

1] λt1p ,+∞[ (|D|−p )|D|−p |f ′ (t)|dt.

There exists a constant C > 0 such that µs (|D|−p ) ≤ C/(s + 1), therefore we have for any t > 0: Z τ (1] λt1p ,+∞[ (|D|−p )|D|−p ) = µs (|D|−p )ds {s>0,µs (|D|−p )> λt1p } Cλtp

C ds s+1 0 C Log(Cλtp + 1).

Z

≤ = We thus deduce from (6) that τ (|f (

|D| )|D|−p )|) ≤ C λ1/p

and hence:

Z

Log(Cλtp + 1)|f ′ (t)|dt,

0

|D| 1 τ (T f ( 1/p )|D|−p ) ≤ kT kC Log(λ + 1) λ 1 −p limλ→ω Log(λ+1) ) τ (T f ( λ|D| 1/p )|D|

+∞

Z

0

+∞

Log(Cλtp + 1) ′ |f (t)dt| ≤ C ′ kT k. Log(λ + 1)

Thus, ST (f ) := makes sense for any f ∈ Cc∞ (R), and defines a linear form ∞ ST on Cc (R). To prove the corollary, we may assume w.l.o.g. that T ≥ 0. In this case, ST is a positive Radon measure with support in [0, +∞[. Moreover, since limλ→ω h(aλ) = limλ→ω h(λ), for any a > 0, we have by Theorem 1: Z 1 τ (T 1]λ−1/p ,+∞ (|D|−1 )|D|−p ) = τω (T |D|−p ), dST = lim (7) λ→ω Log(λ + 1) [0,1[ and the scale invariance of ST implies that Supp(ST ) = {0}. By (7) again, we get: ST = τω (T |D|−p )δ0 ,

14


and the proof is complete. From Rthe above theorem, we deduce the following regularized spectral formula for the noncommutative integral ω T :

Theorem 3. Let (A, M, D) be a (p, ∞)−summable von Neumann spectral triple with p > 1 and D invertible. Let f : [0, +∞) → R be a continuous polynomially decreasing function. Then, for any T ∈ M, the function λ 7→ λ1 τ (T f (λ−1/p |D|)) is well defined and bounded for λ large enough, and we have: Z τ (T f ( λ|D| 1/p ) Cp (f ) ω T = lim , λ→ω λ R +∞ where Cp (f ) = p 0 f (t)tp−1 dt. This formula was proved by A. Connes [16] for M = B(H).

Proof. Let g, h be the functions defined on [0, +∞[ by: Z +∞ Z +∞ ds p = f (s)sp−1 ds. g(t) = t f (t) and h(t) = g(s) s t t

These functions are then continuous on [0, +∞[ and vanish at infinity. Moreover, h′ (t) = −f (t)tp−1 so that R +∞ h is a C 1 function satisfying 1 |h′ (t)| Log(t)dt < +∞. τ (T f (

|D|

)

λ1/p Let us first prove that the function λ 7→ is well defined and bounded for λ > 0. We set for any λ t > 0, ϕ(t) = f (1/t). We thus define a non decreasing continuous function such that ϕ(0) = 0, and hence: Z τ (f ( λ|D| τ (ϕ(λ1/p |D|−1 )) 1 +∞ 1/p )) = = ϕ(λ1/p µs (|D|−1 ))ds. (8) λ λ λ 0

But there exists a constant C > 0 such that µs (|D|−1 ) ≤ τ (f ( λ|D| 1 1/p )) ≤ λ λ

Z

+∞

ϕ C 0

λ s+1

1/p

C , (s+1)1/p

!

where the last equality follows from the change of variable bounded, and the inequality | shows that the function λ 7→ x ≥ 0:

τ (T f (

|D|

))

+∞

f (t)tp−1 C p dt,

C −1 λ−1/p

λ 1/p = C( s+1 ) . We deduce that λ 7→

τ (f (

|D| λ1/p

λ

)

is

is well defined and bounded. On the other hand we have for any

i h p 1 h(x/λ1/p )x−p − h(x)x−p = Log(λ) Log(λ1/p )

and hence

1 t

Z

τ (T f ( λ|D| τ (f ( λ|D| 1/p )) 1/p )) | ≤ kT k λ λ

λ1/p

λ

ds = p

and since ϕ is non decreasing, we get:

Z

1

λ1/p

g(x/s)x−p

ds 1 = s Log(λ)

Z

λ 1

1 x dt f 1/p , t t t

Z λ |D| |D| dt 1 1 p −p −p Th T f ( 1/p ) . = |D| − T h(|D|)|D| 1/p Log(λ) Log(λ) 1 t t λ t By using the equality 8, it is easy to check that the right hand side is a Bochner integral of continuous functions with values in L1 (M, τ ). On the other hand, τ is a continuous form on L1 (M, τ ) and so: Z λ p 1 |D| dt −p −p τ (T h |D| ) − τ (T h(|D|)|D| ) = θ(t) , 1/p Log(λ) Log(λ) t λ 1


15

−p )). Since θ is bounded, we get: with θ(t) = 1t τ (T f ( t|D| 1/p )|D| −p τ (T h( λ|D| ) 1/p )|D| = lim M (θ)(λ) = lim θ(λ). λ→ω λ→ω λ→ω Log(λ) Now by Lemma 1, the left hand side is equal to Z +∞ ph(0)τω (T |D|−p ) = p f (t)tp−1 dt τω (T |D|−p ).

p lim

0

We thus get:

Cp (f )τω (T |D|−p ) = lim

λ→ω

|D| 1 τ (T f ( 1/p )), λ λ

and the proof is complete.

Question. Is the above theorem true for a limiting process ω(f ) = limt→ω f (t) satisfying only the scale invariance property? We can now deduce the Weil formula: Corollary 1. Let (A, M, D) be a p−summable von Neumann spectral triple with p > 1 and D invertible. For any T ∈ M, we have: Z p p −t2 D2 + 1) lim t τ (T e ) = Γ( ω T. 2 t−p →ω 2

Proof. Take f (x) = e−x in the previous theorem.

2.3. Index theory in von Neumann algebras. As before, let M be a von Neumann algebra in a Hilbert space H, equipped with a semi-finite normal faithful trace τ . Lemma 2. For any τ -compact projection e ∈ M, we have τ (e) < +∞. Proof. Since e = e∗ = e2 , we have µt (e) ∈ {0, 1}. But µt (e) → 0 as t → +∞ by hypothesis, so that there exists t0 such that µt (e) = 0, for t ≥ t0 , R t0 and hence τ (e) = 0 µt (e)dt < +∞.

Definition 6. An operator T ∈ M is called τ -Fredholm if there exists S ∈ M such that 1 − ST and 1 − T S are τ -compact.

Proposition 3. If T ∈ M is τ -Fredholm, then the kernel and cokernel projections pT and pT ∗ are τ -finite. Proof. Let S be as in Definition 6. The projections pT = (1−ST )pT and pT ∗ = (1−T S)∗pT ∗ are τ -compact, and Lemma 2 gives the result. Definition 7. The index Indτ (T ) of a τ -Fredholm operator T is defined by: Indτ (T ) := τ (pT ) − τ (pT ∗ ),

(9)

where pT and pT ∗ are the projections on the kernel of T and T ∗ respectively. Proposition 4. If T and S are τ -Fredholm operators, then ST is a τ -Fredholm operator and Indτ (ST ) = Indτ (T ) + Indτ (S); ′

′

Proof. If T and S are parametrices for T and S respectively, then T ′ S ′ is a parametrix for ST . So the composite of two τ -Fredholm operators is a τ -Fredholm operator. In addition: Indτ (ST ) = τ (pST ) − τ (pT ∗ S ∗ ) = τ (pT ) + τ (prKer(ST )⊖Ker(T ) ) − τ (pS ∗ ) − τ (prKer(T ∗ S ∗ )⊖Ker(S ∗ ) ) = τ (pT ) + τ (prIm(T )∩Ker(S) ) − τ (pS ∗ ) − τ (prIm(S ∗ )∩Ker(T ∗ ) ) = Indτ (T ) + Indτ (S).

16


Let us mention a technical lemma which will be used in the sequel. Lemma 3. Let M be a semi-finite von Neumann algebra acting on a Hilbert space H, and let τ be a (positive) normal faithful semi-finite trace on M. Let e, f be two (orthogonal) projections in M and A, B ∈ L1 (M, τ ). Assume that A ∈ eMe, B ∈ f Mf and that there exists V ∈ f Me such that: (i) V A = BV ; (ii) V : e(H) → f (H) is injective with dense range. Then τ (A) = τ (B). Proof. For any ǫ > 0, let pǫ be the spectral projection of |V | corresponding to the interval (ǫ, +∞). Since we have kV pǫ xk ≤ kV xk, for any x ∈ H, the map V x 7→ V pǫ x extends to a contraction qǫ ∈ B(f (H)). Set qǫ x = 0 for x ∈ (1 − f )(H). We thus define an operator qǫ ∈ f Mf which satisfies by construction: qǫ V = V pǫ . qǫ2 V

From this relation, we deduce that = qǫ V and hence qǫ2 = qǫ on f (H). It follows that qǫ2 = qǫ and, since qǫ is a contraction, it is an orthogonal projection in M such that qǫ ≤ f . Moreover, we have: qǫ Bqǫ V = qǫ BV pǫ = qǫ V Apǫ = V pǫ Apǫ . From the inequality: kV xk2 =< |V |2 pǫ x, pǫ x > ≥ ǫ2 kpǫ xk2 = ǫ2 kxk2 , for any x ∈ pǫ (H), we get the existence of an inverse W : qǫ (H) → pǫ (H) for V : pǫ (H) → qǫ (H), such that W ∈ pǫ Mqǫ . We have W qǫ Bqǫ V = pǫ Apǫ and hence, since B ∈ L1 (M, τ ): (10)

τ (qǫ Bqǫ ) = τ (W qǫ Bqǫ V ) = τ (pǫ Apǫ ).

Since V is injective, pǫ → e strongly when ǫ → 0, and it follows from the relation qǫ V = V pǫ that qǫ → f strongly when ǫ → 0. By the Lebesgue dominated convergence theorem in L1 (M, τ ), we deduce from Equation (10) that: τ (eAe) = τ (f Bf ), and finally τ (A) = τ (B). The proof is complete.

The following proposition generalizes the Calderon formula and computes Indτ (T ) by using the powers of 1 − ST and 1 − T S. This formula will be used to get a polynomial expression for the Chern-Connes character (see 2.4). Proposition 5. Let M be a semi-finite von Neumann algebra with a (positive) normal faithful semi-finite trace τ and T ∈ M. Assume that there exists p ≥ 1 and an operator S ∈ M such that: 1 − ST ∈ Lp (M, τ ) and 1 − T S ∈ Lp (M, τ ). Then T is τ -Fredholm and we have for any integer n ≥ p: (11)

Indτ (T ) = τ [(1 − ST )n ] − τ [(1 − T S)n ] .

Proof. The operator T is τ -Fredholm because Lp (M, τ ) ∩ M ⊂ K(M, τ ). To prove the proposition, we may assume that n = 1. Indeed, let S ∈ M be such that: A = 1 − ST and B = 1 − T S are in Lp (M, τ ) ∩ M (and hence in Ln (M, τ ) for any n ≥ p) and set: S ′ = S(1 + B + B 2 + ... + B n−1 ). We have: 1 − T S ′ = B n and 1 − S ′ T = An , where the first relation is immediate and the second one uses the equality T A = BT . Replacing S by S ′ , A by An ∈ L1 (M, τ ) and B by B n ∈ L1 (M, τ ), we are thus reduced to the case where n = 1.


17

When A = 1 − ST and B = 1 − T S are in L1 (M, τ ), we get from the relations ApT = pT and pT ∗ B = pT ∗ the equality: Indτ (T ) = τ (ApT ) − τ (pT ∗ B). To prove that Indτ (T ) = τ (A) − τ (B), it thus suffices to show that: τ (eAe) = τ (f Bf ), where e = 1 − pT and f = 1 − pT ∗ . To this end, set V := f T e. We clearly have: T eA = T A = BT = Bf T, and hence: V (eAe) = (f Bf )V. If the intertwining operator V from e(H) to f (H) were invertible (the inverse would then be automatically in eMf by the bicommutant von Neumann theorem), we would get by cyclicity of the trace: (12)

τ (eAe) = τ (f Bf ),

and Calderon’s formula would be proved. Although V is not necessarily invertible here, it is injective with dense range from e(H) to f (H). It turns out that this is enough to prove (12), by Lemma 3, and therefore Proposition 5 is proved. 2.4. The index map associated with a spectral triple. In this subsection, we describe the index map IndD,τ associated with a p-dimensional von Neumann spectral triple (A, M, D). As usually, we shall replace D by sgn(D), the sign of D. Let D = F1 |D| be the polar decomposition of the self-adjoint operator D. To get rid of the possible non injectivity of F1 and following [15], we replace the Hilbert space H = Ker(D)⊥ ⊕ Ker(D) by (13)

H = H ⊕ Ker(D) ≃ H1 ⊕ H2 ⊕ H3 , ⊥

where H1 = Ker(D) , H2 = Ker(D) and H3 is an extra copy of Ker(D). projections onto H1 , H2 and H3 respectively. According to the splitting (13)    F1 0 0 0 0 F =  0 0 1  = F1 + V where V =  0 0 0 1 0 0 1 We thus define F, V in the semi-finite von Neumann algebra ˜ = M ⊕ pD MpD ⊂ B(H), M

Denote by e1 , e2 and e3 the of H, set:  0 1  0

˜ by a → a ⊕ 0. We have: which is equipped with the trace τ˜ = τ ⊕ τ . Finally, embed A in M Lemma 4. (1) F = F ∗ and F 2 = 1; ˜ τ˜); (2) ∀a ∈ A, [F, a] ∈ Lp,∞ (M, (3) ∀a ∈ A, aF a = aF1 a; ˜ τ˜). (4) ∀a ∈ A,a(F − F1 ) ∈ Lp,∞ (M, Proof. (1) Trivial. (2) Note first that apD (and hence pD a) belongs to Lp,∞ (M, τ ), for any a ∈ A. Indeed, we have: apD = a(D + i)−1 (D + i)pD = ia(D + i)−1 pD ∈ Lp,∞ (M, τ ). On the other hand we have [F1 , a] ∈ Lp,∞ (M, τ ) for any a ∈ A. Indeed we get by easy computations: [F1 , a] = = =

[F1 , a](F1 + pD )(F1 + pD ) [F1 , a]F1 (F1 + pD ) + [F1 , a]pD (F1 + pD ) [F1 , a]F1 (F1 + pD ) + F1 apD (F1 + pD ),

18


where F1 apD (F1 + pD ) ∈ Lp,∞ (M, τ ) by the previous observation. Moreover, we have: [F1 , a]F1 = [D, a](D + i)−1 + i[F1 , a]F1 (D + i)−1 − F1 [|D|, a](D + i)−1 , and hence [F1 , a]F1 ∈ Lp,∞ (M, τ ). It follows that [F1 , a]F1 (F1 + pD ) ∈ Lp,∞ (M, τ ) and finally that: [F1 , a] ∈ Lp,∞ (M, τ )

for any a ∈ A,

as allowed. But we have: [F, a] = [F1 , a] + [V, a], with [V, a] = V pD a(1 − e3 ) − (1 − e3 )apD V e3 . Since [F1 , a], apD and pD a are in Lp,∞ (M, τ ), we finally get: ˜ τ˜), [F, a] ∈ Lp,∞ (M, and (2) is proved. (3) Obvious. (4) We have a(F − F1 ) = aV and hence aV (aV )∗ = aV V ∗ a∗ = apD a. It follows that: µs (aV ) = µs ((aV )(aV )∗ )1/2 = µs ((apD )(apD )∗ )1/2 = µs (apD ), for any s > 0, and the result follows since we know that apD ∈ Lp,∞ (M, τ ).

We are now in position to define the index map IndD,τ : K∗ (A) −→ R, associated with any von Neumann spectral triple (A, M, D). The even case. Assume that the spectral (A, M, D) is even and denote by γ ∈ M the grading involution on M. For any self-adjoint idempotent e ∈ Mn (A), the operator: T = e ◦ (F ⊗ 1n ) ◦ e = e ◦ (F1 ⊗ 1n ) ◦ e anticommutes with γ and satisfies: T 2 − e = e ◦ [F ⊗ 1n , e] ◦ (F ⊗ 1n ) ◦ e = e ◦ [F ⊗ 1n , e] ◦ [F ⊗ 1n , e]. ˜ ⊗ Mn (C))e, τ˜ ⊗ Tr) and hence T is a (˜ It follows that T 2 − e ∈ Lp/2,∞ (e(M τ ⊗ Tr)-Fredholm operator in the ˜ ⊗ Mn (C))e acting on e(Hn ). von Neumann algebra e(M n n ). ) to e(H− Denote by IndD,τ (e) the (˜ τ ⊗ Tr)-index of the positive part of T acting from e(H+ ′ ′ If e, e are two self-adjoint idempotents representing a class [e]−[e ] in K0 (A), then the number IndD,τ (e)− IndD,τ (e′ ) only depends on the class of [e] − [e′ ] in the even K−theory group K0 (A) of the algebra A. The τ -index map thus induces an additive map: (14)

IndD,τ : K0 (A) −→ R. The odd case. The construction of IndD,τ for an odd von Neumann spectral triple (A, M, D) is closely related to the Atiyah-Lusztig spectral flow in the context of von Neumann algebras [11]. Assume for simplicity that A is unital and let P = (1 + F )/2 be the Szego projection associated with the symmetry F . Consider for any invertible matrix u ∈ GLN (A), the Toeplitz operator: T := (P ⊗ 1N ) ◦ u ◦ (P ⊗ 1N ). Since we have: (15)

(P ⊗ 1N ) ◦ u−1 ◦ (P ⊗ 1N ) ◦ u ◦ (P ⊗ 1N ) − (P ⊗ 1N ) = ˜ ⊗ End(CN )), (1/4)[F ⊗ 1N , u−1 ] ◦ [F ⊗ 1N , u] ◦ (P ⊗ 1N ) ∈ Lp/2,∞ (M

and (16) (P ⊗ 1N ) ◦ u ◦ (P ⊗ 1N ) ◦ u−1 ◦ (P ⊗ 1N ) − (P ⊗ 1N ) = (1/4)[F ⊗ 1N , u] ◦ [F ⊗ 1N , u−1 ] ◦ (P ⊗ 1N ),


19

we deduce that T is a (τ ⊗ Tr)-Fredholm operator. Denote by IndD,τ (u) the (τ ⊗ Tr)-index of T . By classical arguments, we again easily see that IndD,τ (u) only depends on the class of u in the odd K−theory group K1 (A) of the algebra A. We get in this way an additive map: IndD,τ : K1 (A) −→ R. To sum up, any von Neumann spectral triple (A, M, D) gives arise to an index map: IndD,τ : K∗ (A) −→ R. This map will be described in Theorem 4 as a pairing with a (polynomial) cyclic cocycle on the algebra A. We shall use the cyclic cohomology of the algebra A and we proceed now to recall the main definitions for the convenience of the reader. Our main references are [17, 15, 31, 41]. Let A be a topological algebra. Denote for k ≥ 0 by C k (A) the vector space of jointly continuous (k + 1)−linear forms on A˜ × Ak+1 , where A˜ is the (minimal) unitalization of A even if A is already unital. The elements of C k (A) are called (continuous) Hochschild cochains on A. The Hochschild coboundary b : C k (A) → C k+1 (A) is the differential defined for any ϕ ∈ C k (A) by: bϕ(˜ a0 , a1 , · · · , ak+1 ) :=

k X (−1)j ϕ(˜ a0 , a1 , · · · , aj aj+1 , aj+2 , · · · , ak+1 )+ j=1

ϕ(˜ a0 a1 , a2 , · · · , ak+1 ) + (−1)k+1 ϕ(ak+1 a ˜0 , a1 , · · · , ak ).

We have b2 = 0. The homology of the resulting complex (C ∗ (A), b) is called the Hochschild cohomology of the algebra A and denoted HH∗ (A). For instance, one can see that 0−dimensional Hochschild cocycles are exactly continuous traces on A. Cyclic cohomology is obtained by using a suitable subcomplex of the Hochschild complex. More precisely, we consider the subspace Cλk (A) of C k (A) built up from jointly continuous (k + 1)−linear forms on A˜ × Ak+1 which are equivariant with respect to the action of the cyclic group generated by the permutation λ(0, 1, · · · , k) = (k, 0, 1, · · · , k − 1). So a Hochschild cochain ϕ is a cyclic cochain if ϕ(ak , a0 , · · · , ak−1 ) = (−1)k ϕ(a0 , · · · , ak ),

∀aj ∈ A.

The Hochschild differential b preserves the subspace Cλ∗ (A) and we get a differential complex (Cλ∗ (A), b) called the cyclic complex of A. Its homology is called the cyclic cohomology of A and is denoted HC∗ (A) or equivalently H∗λ (A). The short exact sequence of complexes 0 → Cλ∗ (A) ֒→ C ∗ (A) −→ C ∗ (A)/Cλ∗ (A) → 0 induces the famous (SBI)−long exact sequence [17] S

I

B

S

I

· · · −→ HCk (A) −→ HHk (A) −→ HCk−1 (A) −→ HCk+1 (A) −→ · · · The operator I is induced by the inclusion and the operator B is defined for instance in [17]. We have B 2 = 0 and bB + Bb = 0. Using the bicomplex (C k,h (A), b, B) with C k,h (A) := C k−h (A) for k ≥ h ≥ 0, we actually recover the cyclic cohomology of A, see for instance [31]. The operator S : HC∗ → HC∗+2 is Connes’ periodic operator, see the definition in [17][page ?]. The homotopy invariants will rather live in periodic cyclic cohomology. This is defined as the strict indutive limit of cyclic cohomology with respect to the operator S and is denoted HP∗ (A). Therefore, HP∗ (A) is a Z2 −graded theory with many topological properties similar to those of K−theory. The main property of cyclic cohomology which will be used in the sequel is its pairing with K−theory. So cyclic cocycles furnish group morphisms from K−theory to the scalars. More precisely, if ϕ is, say, a

20


(2k)−cyclic cocycle on A, then the following formula induces a pairing with the K0 −group of A, see [17]: < ϕ, e >:=

N X

ϕ(ei0 i1 , ei1 i2 , · · · , ei2k i0 ),

e ∈ MN (A), e2 = e.

i0 ,··· ,i2k =1

In the same way, any odd cyclic cocycle induces a pairing with the K1 −theory and a similar explicit formula holds in the odd case. Remark 2. The above definitions of the continuous cyclic cohomology can be generalized to algebras with topologies which are not necessarily topological algebras. Such algebras are then allowed to satisfy some weaker assumptions, see for instance [4, 5] or [21]. Let φ be a cyclic k-cocycle on the algebra A. As in [17], we shall denote for any N ≥ 1, by φ♯ Tr the cyclic k-cocycle on MN (A) given by: (φ♯ Tr)(a0 ⊗ A0 , · · · , ak ⊗ Ak ) := φ(a0 , · · · , ak ) Tr(A0 · · · Ak ), for any (a0 , · · · , ak ) ∈ Ak+1 and any (A0 , · · · , Ak ) ∈ MN (C)k+1 . Theorem 4. Let (A, M, D) be a von Neumann-spectral triple of dimension p and let F be the symmetry associated with D as above. (1) If (A, M, D) is even with grading involution γ, the formula: φ2k (a0 , ..., a2k ) = (−1)k τ (γa0 [F, a1 ] · · · [F, a2k ]); defines, for any k > p/2, a 2k-cyclic cocycle on the algebra A and we have for any projection e in MN (A): IndD,τ (e) = (φ2k ♯ Tr)(e, · · · , e). (2) If (A, M, D) is odd, then for any k > p/2, we define a 2k + 1-cyclic cocycle on the algebra A by setting: φ2k+1 (a0 , ..., a2k+1 ) = (−1/22k+1 )τ (a0 [F, a1 ]...[F, a2k+1 ]); Assume that A is unital, then for any invertible u in MN (A), we have: IndD,τ (u) = (φ2k+1 ♯ Tr)(u−1 , u, ..., u−1 , u). Proof. The proof follows the lines of [17]. (1) We first point out that φ2k is in evidence a cyclic cochain. Let (a0 , · · · , a2k+1 ) ∈ A2k+2 . Then we have: b(φ2k )(a0 , · · · , a2k+1 ) = (−1)k τ (γ[a0 [F, a1 ] · · · [F, a2k ], a2k+1 ]) = (−1)k τ ([a0 [F, a1 ] · · · [F, a2k ], γa2k+1 ]) = 0. Therefore the cochain φ2k is a Hochschild cocycle on A. From Equation (14), we deduce that the operator T = (e ◦ (F ⊗ idN ) ◦ e)+ is τ -Fredholm in e(M ⊗ MN (C))e with parametrix given by S = (e ◦ (F ⊗ idN ) ◦ e)− . Moreover, e − ST as well as e − T S are in Lk (e(M ⊗ MN (C))e, τ ⊗ Tr). Therefore Proposition 5 gives: Indτ ((eF e)+ ) = (τ ♯ Tr)(γ ◦ (e − (e ◦ (F ⊗ idN ) ◦ e)2 )k ). Computing (e − (e ◦ (F ⊗ idN ) ◦ e)2 )k and using the relation e ◦ [F ⊗ 1N , e] ◦ e = 0, we obtain: (e − (e ◦ (F ⊗ idN ) ◦ e)2 )k = (−1)k e ◦ [F ⊗ 1N , e]2k , and hence the conclusion. (2) That φ2k+1 is cyclic is again obvious. Let (a0 , · · · , a2k+2 ) ∈ A2k+3 . Then we have: b(φ2k+1 )(a0 , · · · , a2k+2 ) = (−1/22k+1 )τ ([a0 [F, a1 ] · · · [F, a2k+1 ], a2k+2 ]) = 0. Hence φ2k+1 is a cylic cocycle on A.


21

To compute the τ -index of P ◦u◦P , we again apply the Calderon formula. From the relations (15) and (16), we deduce that T := P ◦ u ◦ P is τ -Fredholm in P (M ⊗ MN (C))P with parametrix given by S = P ◦ u−1 ◦ P . Moreover, P − ST and P − T S are in Lk (P (M ⊗ End(CN ))P, τ ⊗ Tr). Therefore Proposition 5 gives: Indτ (P ◦ u ◦ P ) = (τ ♯ Tr)((P − (P ◦ u−1 ◦ P ◦ u ◦ P ))k ) − (τ ♯ Tr)((P − (P ◦ u ◦ P ◦ u−1 ◦ P ))k ). The computation of P − (P ◦ u−1 ◦ P ◦ u ◦ P ) in (15) and (16) gives: P − (P ◦ u−1 ◦ P ◦ u ◦ P ) = −[P, u−1 ] ◦ [P, u] ◦ P and P − (P ◦ u ◦ P ◦ u−1 ◦ P ) = −[P, u] ◦ [P, u−1 ] ◦ P. But, ([P, u−1 ] ◦ [P, u] ◦ P )k = ([P, u−1 ] ◦ [P, u])k ◦ P and ([P, u] ◦ [P, u−1 ] ◦ P )k = ([P, u] ◦ [P, u−1 ])k ◦ P. On the other hand we have: [P, u−1 ] ◦ [P, u] = −u−1 ◦ [P, u] ◦ u−1 ◦ [P, u] and [P, u] ◦ [P, u−1 ] = −u ◦ [P, u−1 ] ◦ u ◦ [P, u−1 ], Therefore, ([P, u−1 ] ◦ [P, u])k = (−1)k (u−1 ◦ [P, u])2k = (−1)k u−1 ◦ ([P, u] ◦ u−1 )2k−1 ◦ [P, u], and a similar result holds for ([P, u] ◦ [P, u−1 ])k and we get: Indτ (P ◦ u ◦ P ) = (−1)k (τ ♯ Tr)(P ◦ u−1 ◦ ([P, u] ◦ u−1 )2k−1 ◦ [P, u] − P ◦ ([P, u] ◦ u−1 )2k−1 ◦ [P, u] ◦ u−1 ). Hence we get using the trace property of τ : Indτ (P ◦ u ◦ P ) = (−1)k (τ ♯ Tr) [P, u−1 ] ◦ ([P, u] ◦ u−1 )2k−1 ◦ [P, u] ,

and thus we finally obtain:

Indτ (P ◦ u ◦ P ) = (−1/22k+1 )(τ ♯ Tr) u−1 ◦ [F ⊗ 1N , u] ◦ ([F ⊗ 1N , u−1 ] ◦ [F ⊗ 1N , u])k ,

which completes the proof.

Remark 3. One can define a cyclic cocycle of minimal order. In the even case for instance, there is a well defined cyclic p−cocycle that can be associated with the spectral triple in the following way: (−1)p/2 τ (γF [F, a0 ] · · · [F, ap ]). 2 This cocycle also represents the index map associated with the spectral triple. The proof is an easy extension of the one given in [17]. φp (a0 , · · · , ap ) :=

Remark 4. Assume that G is a compact Lie group which acts on the even spectral triple [3]. So G acts by unitaries in M, this action preserves A and the operator D is G-invariant. We denote by U (g) the unitary corresponding to g ∈ G. Then the equivariant index IndG τ ((eF e)+ ) of (eF e)+ does make sense as an element of R(G) ⊗ R, where R(G) is the representation ring of G. We get using a similar proof the following equivariant polynomial index formula: ∀g ∈ G, IndG τ ((eF e)+ )(g) = (φ2k ♯ Trace)(U (g) ◦ e, e, ..., e). See [3]. A similar result holds in the odd case. So associated with any von Neumann spectral triple, there is an index problem which can be stated as follows: ”Give a local formula for the traced index map K∗ (A) −→ R.” Using Theorem 4, we see that the index problem can be stated in the cyclic cohomology world: ”Find a local cyclic cocycle ψ on A such that: < ψ, x >=< φ, x >,

∀x ∈ K∗ (A).′′

22


Here φ is the cyclic cocycle defined in Theorem 4. This index problem reduces to the index problem solved by A. Connes and H. Moscovici in [19] if one takes the usual von Neumann algebra of operators in a Hilbert space with the usual trace. Examples. In the examples listed after Definition 5, the index problem becomes: (1) In the first example of Riemannian geometry, we recover the classical index problem which was solved by Atiyah and Singer in [2, 15]. (2) In the case of measured foliations we recover the measured index problem which was solved by A. Connes in [13]. (3) In the case of Galois coverings, we recover the index problem which was solved by M.F Atiyah in [1]. (4) For almost periodic operators, we obtain the Shubin index problem that was solved in [39]. The index map yields here a morphism: IndD,τ : K 0 (RnB ) −→ R, n where RB is the Bohr compactification of Rn . Up to normalizing constants, the sequence φn of Theorem 4 can be arranged to represent a periodic cyclic cocycle on A [17], i.e. up to appropriate constants, we have: S(φn ) = φn+2 , where S is Connes’ periodic operator. The periodic cyclic class obtained is called the Chern-Connes character of the von Neumann spectral triple. In [6] we give a local formula for this Chern-Connes character using residues of zeta functions and following the method of [19]. This local formula unifies all the examples listed above and gives a complete solution to the von Neumann index problem. Remark 5. Any even von Neumann spectral triple gives rise for n large enough to a homomorphism from the Cuntz algebra qA of A to the algebra J = L2n (M, τ ) [18]. This shows that the Chern-Connes character of the von Neumann spectral triple can also be defined following the method of [18]. 3. Measured foliations Let (M, F ) be a smooth foliated manifold with (for simplicity) even dimensional spin leaves. Denote by G the holonomy groupoid of (M, F ). Let S be the hermitian spin vector bundle and D the G−operator constructed out of the Dirac operator along the leaves following [13]. For all the background material about G−operators, we refer to the seminal paper [13]. We fix a Lebesgue-class measure α on the leaf manifold F and the lifted Haar system ν = (νx )x∈M on G. We assume furthermore that there exists a positive holonomy invariant transverse measure Λ, then the data (Λ, α) enables to define a measure on the manifold M that we denote by Λν [13]. We will denote by WΛ∗ (M, F ; S) the von Neumann algebra associated with Λ and S [6]. This von Neumann algebra is then endowed with a trace τΛ whichRturns out to be faithful by construction. ⊕ Recall that WΛ∗ (M, F ; S) is represented in the Hilbert space H = M L2 (Gx , s∗ (S), ν x )dΛν (x) of Λν -square 2 x ∗ integrable sections of the field of Hilbert spaces (Hx = L (G , s (S), ν x ))x∈M , where Λν is the positive measure on M constructed out of Λ and ν as in [13]. Definition 8. Let (M, F, Λ) be a measured p−dimensional foliation on a compact manifold M . For any pseudodifferential G-operator P of order −p acting on sections of a vector bundle E over M , we define the foliated local residue res(P ) ∈ C ∞,0 (M, |Λ|1 F ) as the longitudinal 1-density, given locally by: "Z # 1 tr(σ−p P ((u, t), ξ))|dξ| |du|, res(u,t) (P ) = (2π)p kξk=1 where σ−p P ((u, t), ξ) is the principal symbol of P . This local residue is well defined (see for instance [30], p.17) and we have the following generalization of the well known result in the non foliated case and which shows the locality of the von Neumann Dixmier


23

trace in the case of measured foliations. Theorem 5. Let (M, F, Λ) be a measured p-dimensional foliation on a compact manifold M . Let P be a pseudodifferential G−operator of order −p acting on sections of a vector bundle E over M , and denote by σ−p (P ) its principal symbol. Then we have (i) P belongs to the Dixmier ideal L1,∞ (WΛ∗ (M, F ; E), τΛ ) associated with the von-Neumann algebra ∗ WΛ (M, F ; E) and its trace τΛ ; (ii) For any invariant mean ω, the Dixmier trace of P is given by Z 1 τωΛ (P ) = resL (PL )dΛ(L), p M/F where resL (PL ) is the foliated local residue of Definition 8 and τωΛ is the Dixmier trace associated with τΛ and ω as in Section 1. Proof. (i) By [15][page 126], we have P =

X

Pi + R

1≤i≤k

where R is an infinitely smoothing G-operator and each Pi ∈ ψ −p (M, F ; E) is given by a continuous familly with compact support in ψc−p (Ωi , E) with Ωi a distinguished foliation chart trivializing E. Since R is traceclass with respect to τ Λ , [13][Prop. 6.b, page 131] and L1 (WΛ∗ (M, F ; E)) ⊂ L1,∞ (WΛ∗ (M, F ; E)) we may assume that P ∈ ψc−p (Ω, E) where Ω is a distinguished foliation chart trivializing E. We thus may work locally assuming that M = T p × Dn−p is foliated by T p × {t}, for t ∈ Dn−p and P = (Pt )t∈Dn−p is a continuous familly of scalar pseudodifferential operators of order −p on T p (the proof for matrices is the same). Here T p is the standard p-torus and Dn−p is the unit disk in Rn−p . For any t ∈ Dn−p , let ∆t = ∆ be the usual Laplacian on the flat torus T p . Since L1,∞ (WΛ∗ (M, F )) is an ideal in WΛ∗ (M, F ), we only have to show that the constant familly (1 + ∆t )−p/2 defines an element in L1,∞ (WΛ∗ (M, F )). Indeed we have Pt = Qt (1 + ∆t )−p/2 ,

(t ∈ Dn−p )

where Qt = Pt (1 + ∆t )p/2 is a continuous family of 0-order pseudodifferential operators on T p , and hence defines an element of WΛ∗ (M, F ) ∼ = L∞ (Dn−p , Λ) ⊗ B(L2 T p ) by [13][ page 126, Proposition 1.b]. Since we trivially have for any T ∈ B(L2 (T p )): Λ

µτΛ(Dn−p )s (1 ⊗ T ) = µTr s (T ), we get: 1 Log(R)

Z

0

R

Λ µτs ((1

−p/2

⊗ (1 + ∆))

Λ(Dn−p ) )ds = Log(R/Λ(Dn−p ))

The right hand side of this equality converges to to L1,∞ (WΛ∗ (M, F )) and we get

Λ(Dn−p ) p

τωΛ (1 ⊗ (1 + ∆)−p/2 ) =

Z

R/Λ(Dn−p )

µs (1 + ∆)ds.

0

× Area(T p ). Henceforth, 1 ⊗ (1 + ∆)−p/2 belongs

Λ(Dn−p ) × Area(T p ). p

(ii) We may work locally and assume again that M = T p ×Dn−p . For any smooth function σ = σ(u, ξ, t) ∈ S T p × Dn−p , set ν(σ) = τωΛ (P ), ∗

24


where P is any classical tangential pseudodifferential operator of order −p with principal symbol equal to σ. Since two classical pseudodifferential operators of order −p with the same principal symbol coincide modulo ψ −(p+1) (M, F ) and since ψ −(p+1) (M, F ) ⊂ L1 (WΛ∗ (M, F )) ⊂ Ker(τωΛ ), we deduce that ν(σ) is well defined. It is clear that ν is a positive linear form on C ∞ (S∗ T p × Dn−p ) and is in fact a positive measure on T ∗ (T p )1 × Dn−p . R Let ν = Dn−p νt dρ(t) be the disintegration of ν with respect to the projection π : S∗ (T p ) × Dn−p → Dn−p [8][page 58]. For any isometry g of T p , the measure ν is invariant under the action of g on the fibers of π because τωΛ is a trace. By uniqueness of the disintegration of ν we get g(νt ) = νt ,

ρ − a.e. in t,

∗

p

so that νt is proportional to the volume form on S T for almost every t. We thus have Z Z σ−p (P )(u, ξ, t)dv(u, ξ)]h(t)dρ(t), [ τωΛ (P ) = Dn−p

S∗ T p

where h is a bounded ρ-measurable positive function on Dn−p . Let us prove now that the measure hdρ is proportional to dΛ. For any continous function f on Dn−p , we know by (i) that the Dixmier trace of the continuous family Pt = f (t)(1 + ∆t )−p/2 where ∆ = ∆t is now the Laplacian on the standard sphere, is given by τωΛ (P ) = C1 × Λ(f ), the constant C1 being independent of f . On the other hand, we have τωΛ (P ) = C2 ×

Z

f (t)h(t)dρ(t),

Dn−p

where the constant C2 does no more depend on f . We thus get the existence of a constant C > 0 such that hdρ = CdΛ. It follows that τωΛ (P ) = C and the computation of (i) shows that C = 1/p.

Z

resL (PL )dΛ(L), M/F

Proposition 6. Let γ be the grading induced by the grading of the spin bundle S (dim(F ) = 2r) and let G be the symmetry constructed out of D like in Section 2 so that G 2 = 1. For any f ∈ C ∞ (M ), denote by π(f ) the 0 order differential G-operator defined by f , say multiplication by f ◦ s on each L2 (Gx , ν x , s∗ S). Then: (i) (C ∞ (M ), W ∗ (M, F ; S), D) is an even von-Neumann spectral triple of finite dimension equal to the dimension of the leaves; (ii) ∀k > r, φk (f0 , f1 , ..., f2k ) = (−1)k (τΛ ♯trace)(γ ◦ π(f0 ) ◦ [G, π(f1 )] ◦ ... ◦ [G, π(f2k )]) defines a cyclic cocycle on the algebra C ∞ (M ); (iii) Let e ∈ MN (C ∞ (M )) be the projection corresponding to a stabilization of the complex vector bundle E. Then we have for any k > r: IndΛ ([DE ]+ ) = φk (e, · · · , e). Proof. We only have to prove (i), the rest of the proposition being a rephrasing of Theorem 4 in the present situation. We first point out that D is affiliated with W ∗ (M, F ; S) and that ∀f ∈ C ∞ (M ), [D, f ] is in W ∗ (M, F ; S) because it is affiliated and bounded. On the other hand the principal symbol of |D| commutes with those of all order 0 pseudodifferential G-operators, so that (ii) and (iii) in Definition 5 are satisfied. Let now Q ∈ ψ −1 (M, F ; S) be a parametrix for the elliptic G-operator D so that 1 − QD = R and 1 − DQ = R′


25

are regularizing operators, say live in Cc∞,0 (G, End(S)). The existence of Q is proved in [13]. Then we have (D + i)−1 = Q + (D + i)−1 R′ so that (D + i)−1 and Q are in the same Dixmier ideal. But Q ∈ L2r,∞ (W ∗ (M, F ; S), τΛ ) and the conclusion follows. For any smooth complex vector bundle E over M , the twisted Dirac operator DE (lifted again to become a G-operator) is a well defined elliptic differential G-operator. The von Neumann index problem then asks for a computation of the measured analytic map K 0 (M ) → R given by: [E] → IndΛ ([DE ]+ ), as a pairing of E with a cyclic cocycle on C ∞ (M ). Whence we joint the usual measured index problem, at least for spin foliations.

4. The local positive Hochschild class In this final section, we shall prove a local formula for the image of the Chern-Connes character of a von Neumann spectral triple in Hochschild cohomology. More precisely, we shall give a local representative of this class in terms of the Dixmier trace associated with some state ω. The formula that we obtain shows at the same time the positivity of the Hochschild cocycle [18, 15]. Our results follow from the classical case treated by A. Connes in the unpublished paper [16] modulo the results of the previous sections. A proof of these technical results in the type I case also appeared in the meantime in [42]. For the sake of simplicity, we shall restrict ourselves to the even case. The main problem is the following. The Chern-Connes character Ch(A, M, D) of an even p-dimensional von Neumann spectral triple (A, M, D) can be described in the (b, B)−bicomplex by a family (ϕ2k )k≥0 such that bϕ2k + Bϕ2k+2 = 0. Then the pairing < Ch(A, M, D), [e] > with projections is given up to normalization by the formula [15][page 271] X 1 (2k)! ϕ2k (e − , e, · · · , e). < [ϕ], [e] >= (−1)k k! 2 k≥0

A solution to the index problem is a precise periodic cyclic cocycle ϕ where each ϕ2k is given by a formula which is local in the sense of Connes-Moscovici, i.e. only involving suitable residues of operator zeta functions. This problem is dealt with in the forthcoming paper [6], and we concentrate here on a local formula for the Hochschild class of Ch(A, M, D). Since the normalized pairing between cyclic cohomology and K−theory is invariant under the operator S, it is natural to determine the dimension of the Chern character, i.e. the greatest n ≥ 1 such that the analytic Chern-Connes character defined in the previous section is not in the range of the S−operation. Since we have Im(S) = Ker(I), where I : HC∗ (A) → HH∗ (A) is the natural forget map, it is important to have a computable local formula for the image ICh(A, M, D) of the Chern-Connes character in Hochschild cohomology. This would enable to prove or disprove that Ch(A, M, D) is or is not in the image of S. The local formula we get here is Z (17) < ICh(A, M, D), a0 ⊗ · · · ⊗ ap >= ω γa0 [D, a1 ] · · · [D, ap ], where ω is any state satisfying the assumptions (1) and (2) fixed in the first section. The RHS of this formula is local since it involves a Dixmier trace, and it is therefore more computable than the LHS. We assume from now on, and in order to avoid unnecessary complications arising from the use of the Green operators, that Ker(D) = {0}, but all the results remain true in the general case with the easy suitable modifications. Moreover, the local formula is stable under bounded perturbations of D. Recall that the analytic Chern-Connes character is a cyclic cohomology class φ over A which can be represented in the lowest dimension p by the cyclic cocycle: φ(a0 , ..., ap ) := τs (γa0 [F, a1 ]...[F, ap ]),

26


where τs is the regularization (enables to win one degree of summability) defined by 1 τ (γF [F, α]), 2 and γ is the grading involution. The proof given in [17] extends easily to our setting and is omitted. We fix from now on a state ω on Cb (R) which vanishes on C0 (R) and satisfies the assumptions fixed in the first section. For simplicity, we shall denote again for any g ∈ Cb (R) by limt→ω g(t) the number ω(g) obtained out of g as a function of t and defined in Equation (1). Hence, in particular, this functional is invariant under dilations. We shall denote by τω the Dixmier trace associated with the trace τ and the state ω, see Section 1 for more precise definitions. τs (α) :=

Theorem 6. The pairing of ICh(M, D) with Hochschild homology coincides with the pairing of a local Hochschild cocycle φ given by the formula: XZ X i i i i i i a0 ⊗ a1 ... ⊗ ap >:= < φ, ω γa0 [D, a1 ]...[D, ap ]. i

i

To be rigorous, this equality holds only up to constant since it depends on the normalizations of the Chern-Connes character and of the pairing (for a coherent choice of the normalizations, see [15]). The proof of Theorem 6 is based on some technical Lemmas that we state first. P Lemma 5. [16] Let ω = i∈I ai0 ⊗ ... ⊗ aip be a Hochschild cycle. Let f be a compactly supported smooth even function such that f (0) = 1. Then X τs ([f (tD), aip ]ai0 [F, ai1 ]...[F, aip−1 ]F ). < ICh(A, M, D), ω >= − lim t→0

i

Proof. Forget i but keep it in mind! Set A = a0 [F, a1 ]...[F, ap ]. The function ft (x) = f (tx) converges simply to 1 when t goes to 0, and by the Lebesgue theorem, we deduce that f (tD) converges weakly to the identity operator. Since τ is a normal trace and γF [F, A] is trace-class, we have: < ICh(A, M, D), ω >= τ (γF [F, A]) = lim τ (f (tD)γF [F, A]). t→0

But, f (tD)F = F f (tD) and since f is an even function, we also have f (tD)γ = γf (tD). Hence we deduce: < ICh(A, M, D), ω >= lim τs (f (tD)A). t→0

1

Now the operator f (tD) belongs to L (M, τ ) for any t > 0 and the operator ap commutes with the grading involution γ, thus: τs (f (tD)A) = τs (ap f (tD)a0 [F, a1 ]...[F, ap−1 ]F ) − τs (f (tD)a0 [F, a1 ]...[F, ap−1 ]ap F ). Set δ = [F,P·] for the derivation induced by F on A and let us apply the operator id ⊗ δ ⊗ ... ⊗ δ to the equality b( i ai0 ⊗ ai1 ⊗ ... ⊗ aip ) = 0. We get: X X aip ai0 [F, ai1 ]...[F, aip−1 ], ai0 [F, ai1 ]...[F, aip−1 ]aip = i

i

which finishes the proof.

Lemma 6. Let f be an even function in Cc∞ (R) which equals 1 in a neighborhood of 0. Then ∀T ∈ M, ∀a ∈ A we have: Z p −p+1 lim τ ([f (t D), a]T |D| ) = −p ω [|D|, a]T. 1 t

→ω

Proof. Since f is even, this lemma only involves |D| and we can assume D ≥ 0. Let us first formally replace [f (tD), a] by f ′ (tD)[tD, a]. We obtain: lim τ ([f (tD), a]T D−p+1 ) = −p lim τ (tD−p+1 f ′ (tD)[D, a]T )

t−p →ω

t

→ω


27

But setting f1 (x) = x−p+1 f ′ (x) if x ≥ 0 and extending f1 to an even function, we obtain a well defined function f1 satisfying the assumptions of Lemma 5. So: lim τ ([f (tD), a]T D−p+1 ) = −p lim tp τ (g(t|D|)[|D|, a]T ).

t−p →ω

t

→ω

and the proof is complete. It remains thus to show that: A(t) = [f (tD), a] − f ′ (tD)[tD, a] belongs to Lp,∞ (M, τ ) and that kA(t)kp,∞ is an O(t). We now use the Fourier transform: Z fˆ(u)[eiutD , a]du. [f (tD), a] = R

But recall that [e

isD

, a] =

R1 0

e

isvD

is[D, a]e

is(1−v)D

dv, and so: Z f ′ (tD)[tD, a] = iufˆ(u)eiutD [tD, a]du. R

Thus A(t) =

Z

R

fˆ(u)

Z

1

eiutsD [iutD, a]eiut(1−s)D − eiutD [iutD, a])dsdu

0

=

Z

fˆ(u)

1

eiutsD [[iutD, a], eiut(1−s)D ]dsdu

0

R

= t2

Z

Z

u2 fˆ(u)

R

Z

0

1

Z

1

(1 − s)eiut(s+(1−s)r)D [D, [D, a]]eiut(1−s)(1−r)D dsdrdu.

0

Note that [D, [D, a]] is bounded since we could assumed D = |D| in this proof. Thus kA(t)kM = O(t2 ). On the other hand, we have: Z +∞ Z +∞ ds µs (A(t)) µs (A(t))µs (D−p+1 )ds ≤ CkT k |τ (A(t)T D−p+1 )| ≤ kT k , (1 + s)(p−1)/p 0 0

where C is some constant. Let us see that there exists a constant K > 0 such that µs (A(t)) = 0,

∀s > K/tp .

To this end, let A > 0 be such that Supp(f ) ⊂ [−A, A] and denote by Et the spectral projection of D−1 corresponding to the interval [t/A, +∞). We have τ (Et ) = |{s > 0, µs (D−1 ) ≥ t/A}|. Since there exists a constant B > 0 such that µs (D−1 ) ≤ B/s1/p , we get τ (Et ) ≤ |{s > 0, s1/p ≤ AB/t}|, and hence τ (Et ) = O(t−1/p ). But, µs (A(t))

≤

µs/2 (f (tD)a − f ′ (tD)[tD, a]) + µs/2 (af (tD))

=

µs/2 (Et (f (tD)a − f ′ (tD)[tD, a])) + µs/2 (af (tD)Et )

≤

kf (tD)a − f ′ (tD)[tD, a]kµs/2 (Et ) + kaf (tD)kµs/2 (Et ).

The first and the last inequalities follow from [26][Lemma 2.5, (v) and (vi), page 276]. Since τ (Et ) = O(t−1/p ), there exists a constant K > 0 such that µs/2 (Et ) = 0 for s > K/tp , and hence µs (A(t)) = 0 for s > K/tp .

28


It follows that |τ (A(t)T D−p+1 )| ≤ CkT kt2

Z

K/tp

0

and hence τ (A(t)T D−p+1 ) → 0 when t → 0.

ds , (1 + s)(p−1)/p

Proposition 7. The pairing ICh(A, M, D) with Hochschild homology is given by XZ X i i i i p−1 . ai0 ⊗ ai1 · · · ⊗ aip >= −p < ICh(A, M, D), ω γ[|D|, ap ]a0 [F, a1 ]...[F, ap−1 ]D i

i

Cc∞ (R)

Proof. Let f be an even function in which equals 1 in a neighborhood of 0 as above. Then Proposition 5 shows that X X lim τs ([f (tD), aip ]ai0 [F, ai1 ]...[F, aip−1 ]F ). ai0 ⊗ ai1 · · · ⊗ aip >= − < ICh(A, M, D), i

i

t→0

Lemma 6 then gives with the bounded operator T = a0 [F, a1 ]...[F, ap−1 ]F γ|D|p−1 :

− lim τs ([f (tD), aip ]ai0 [F, ai1 ] · · · [F, aip−1 ]F ) = − lim τ ([f (tD), aip ]ai0 [F, ai1 ] · · · [F, aip−1 ]F γ|D|p−1 |D|−p+1 ) t→0 t→0 Z Z i i i i = p ω [|D|, ap ]a0 [F, a1 ] · · · [F, ap−1 ]F γ|D|p−1 = −p ω [|D|, aip ]ai0 [F, ai1 ] · · · [F, aip−1 ]γDp−1 Z = −p ω γ[|D|, aip ]ai0 [F, ai1 ] · · · [F, aip−1 ]Dp−1 .

This completes the proof.

Let δ be the derivation on the algebra A˜ generated by A and [|D|, A] given by δ(X) = [|D|, X]. Recall that for any Hochschild cocycle ρ on B, one defines a new Hochschild cocycle by setting [32]: < δj (ρ), (X0 , X1 , · · · , Xp ) >= ρˆ(X0 dX1 · · · dXj−1 δ(Xj )dXj+1 · · · dXp ), where ρˆ is the corresponding non commutative current [17]. Moreover, it is well known and easy to check that δj (ρ) + δj+1 (ρ) is a Hochschild coboundary. Therefore, the derivation δ induces, by considering the class of (−1)j δj (ρ) for any j ∈ {1, · · · , p}, a map between Hochschild cohomology: ˜ −→ HH∗+1 (A). ˜ iδ : HH∗ (A) Moreover, this map satisfies the relation i2δ = 0 in Hochschild cohomology. Proposition 8. (1) Recall that p is even. We define Hochschild cocycles on A by setting Φ(a0 , · · · , ap ) = τω (γa0 [D, a1 ] · · · [D, ap ]D−p ), Ψ(a0 , · · · , ap ) = pτω (γa0 [F, a1 ] · · · [F, ap−1 ][|D|, ap ]D−1 ) and ϕk (a0 , · · · , ap−1 ) = τω (γa0 [F, a1 ] · · · [F, ak ]Dp−k ), k = 0 · · · p. Moreover, ϕp = 0. (2) The Hochschild cocycles Φ and Ψ are cohomologous in Hochschild cohomology. Proof. (1) We have by straightforward computation: bΦ(a0 , · · · , ap+1 ) = (−1)p τω (γa0 [D, a1 ] · · · [D, ap ][ap+1 , D−p ]) . But [ap+1 , D−p ] belongs to the ideal L1 (M, τ ), since we have:

[ap+1 , |D|−1 ] = −|D|−1 [ap+1 , |D|]|D|−1 , and [ap+1 , D−p ] = j |D|−j [ap+1 , |D|−1 ]|D|−(p−j−1) . In the same way, we have: P

bΨ(a0 , · · · , ap+1 ) = −p(−1)p τω (γa0 [F, a1 ] · · · [F, ap−1 ][|D|, ap ][ap+1 , D−1 ]).


29

But again, [ap+1 , D−1 ] belongs to Lp (M, τ ) and thus the operator γa0 [F, a1 ] · · · [F, ap−1 ][|D|, ap ][ap+1 , D−1 ] is trace class. Therefore bΨ = 0. Thus, bΦ = 0. The proof for the cochains ϕk is similar. Now we obviously have: τω (γa0 [F, a1 ] · · · [F, ap ]) = τω (γF [F, a0 ][F, a1 ] · · · [F, ap ]) = 0. (2) We first point out that [|D|−1 , [D, a]] = −|D|−1 [|D|, [D, a]]|D|−1 . Therefore, in the expression of Φ(a0 , · · · , ap ) we can move |D|−1 to the left and in particular: Φ(a0 , · · · , ap ) = τω (γa0 [D, a1 ]|D|−1 · · · [D, ap ]|D|−1 ).

(18) On the other hand,

[D, a]|D|−1 = [F, a] + F [|D|, a]|D|−1 = [F, a] + [|D|, a]D−1 + [F, [|D|, a]]|D|−1 . Now, since F = D|D|−1 , we deduce [F, [|D|, a]] = [D, [|D|, a]]|D|−1 − F [|D|, [|D|, a]]|D|−1 . Therefore and since [D, [|D|, a]] = [|D|, [D, a]] is bounded, the operator [F, [|D|, a]] belongs to Lp,∞ (M, τ ). Therefore, [D, a]|D|−1 − ([F, a] + [|D|, a]D−1 )

∈ Lp (M, τ ).

Hence we can replace [D, a]|D|−1 by [F, a] + [|D|, a]D−1 when necessary in the expression of Φ in Equation (18). On the other hand, we have: δj (ϕ)(a0 , · · · , ap ) = τω (γa0 [F, a1 ] · · · [F, aj−1 ][|D|, aj ][F, aj+1 ] · · · [F, ap ]D−1 ). Thus and since (−1)j δj (ϕ) is cohomologous to δp (ϕ) = Φ, we deduce that: Φ(a0 , · · · , ap ) =

p X

τω (γa0 [F, a1 ] · · · [F, aj−1 ][|D|, aj ]D−1 [F, aj+1 ] · · · [F, ap ]) + Rp (a0 , · · · , ap ),

j=1

0

where Rp (a , · · · , ap ) corresponds to the terms where the factor [|D|, a]D−1 appears at least twice. The first remark is that D−1 [F, a] + [F, a]D−1 = |D|−1 δ 2 (a)|D|−2 − D−1 δ([D, a])|D|−2 .

(19)

Therefore, we have: τω (γa0 [F, a1 ] · · · [F, aj−1 ][|D|, aj ]D−1 [F, aj+1 ] · · · [F, ap ]) = (−1)j τω (γa0 [F, a1 ] · · · [F, aj−1 ][|D|, aj ][F, aj+1 ] · · · [F, ap ]D−1 ). This latter is nothing but a representative for iδ (ϕ)(a0 , · · · , ap ) for any j. Therefore: Φ(a0 , · · · , ap ) = pτω (γa0 [F, a1 ] · · · [F, ap−1 ][|D|, ap ]D−1 ) + Rp (a0 , · · · , ap ) + bα(a0 , · · · , ap ), for some cochain α. To finish the proof, we thus need to show that Rp is a coboundary. But this is a consequence of the fact that i2δ = 0 in Hochschild cohomology. More precisely, consider for instance the Hochschild cocycle ϕ2 (a0 , · · · , ap−2 ) = τω (γa0 [F, a1 ] · · · [F, ap−2 ]D−2 ). Then i2δ ϕ can be represented for 1 ≤ j < k ≤ p by the Hochschild cocycle (a0 , · · · , ap ) 7−→ (−1)jk τω (γa0 [F, a1 ] · · · [F, aj−1 ][|D|, aj ][F, aj+1 ] · · · [F, ak−1 ][|D|, ak ][F, ak+1 ] · · · [F, ap ]D−2 ). But again, this is precisely, τω (γa0 [F, a1 ] · · · [F, aj−1 ][|D|, aj ]D−1 [F, aj+1 ] · · · [F, ak−1 ][|D|, ak ]D−1 [F, ak+1 ] · · · [F, ap ]),

30


and hence corresponds to the (j, k)−term in the expression of Rp . Thus all the terms where [|D|, a]D−1 appears twice are coboundaries. The same argument using the Hochschild cocycles (ϕk )k≥3 shows that all the other terms in Rp are coboundaries. The proof is thus complete. Proof of Theorem 6. By using Proposition 7 and Proposition 8, it is sufficient to show that τω (γa0 [F, a1 ] · · · [F, ap−1 ][|D|, ap ]D−1 ) = −τω (γ[|D|, ap ]a0 [F, a1 ] · · · [F, ap−1 ]D−1 ). But γ[|D|, ap ] + [|D|, ap ]γ = 0, and hence τω (γ[|D|, ap ]a0 [F, a1 ] · · · [F, ap−1 ]D−1 ) = −τω ([|D|, ap ]γa0 [F, a1 ] · · · [F, ap−1 ]D−1 ). Therefore and using the trace property of τω , we get: τω (γ[|D|, ap ]a0 [F, a1 ] · · · [F, ap−1 ]D−1 ) = −τω (γa0 [F, a1 ] · · · [F, ap−1 ]D−1 [|D|, ap ]). Again we can use Equation (19) to move D−1 to the right and this completes the proof. Remark 6. The equivariant case with respect to an action of a compact Lie group can be handled in the same way using for instance the definitions of [3]. The local formula obtained for the equivariant Hochschild chern character is then interesting in view of fixed point theorems because it gives in the usual situations such as compact manifolds and compact foliated manifolds a measure on the cosphere or longitudinal cosphere bundle which is supported by the fixed points. Remark 7. In many interesting situations (e.g.: measured foliations, coverings, almost periodic operators), the local expression of the Hochschild class of the Chern-Connes character obtained in Theorem 6 actually defines a cyclic cocycle on A. If we define the von Neumann Yang-Mills action YMτ (∇) associated with any compatible connection ∇ on a finitely generated projective module Ω over the ∗-algebra A, then we can state following Connes’ method (See [15], Theorem 4, page 561): Proposition 9. Let (A, M, D) be a four dimensional spectral triple. Set φτω (a0 , a1 , a2 , a3 , a4 ) = 2τω (γa0 [D, a1 ][D, a2 ][D, a3 ][D, a4 ]D−4 ) and assume that φτω is cyclic. Then for any hermitian finitely generated projective module Ω over A with a compatible connection ∇, we have | < [Ω], φτω > | ≤ YMτ (∇). The consequences of this proposition in the examples listed before will be treated in a forthcoming paper. We point out that the case of almost periodic operators is especially interesting for applications to quasiperiodic tillings. In the example of measured foliated manifolds, the representative of the Hochschild class of the Chern-Connes character given above, is a cyclic cocycle over C ∞ (M ). The computations in [6] give the following expected result: Theorem 7. The current C in M which represents the Hochschild class of the Chern-Connes character of the Dirac operator along the leaves via the Connes-Hochschild-Kostant-Rosenberg isomorphism is, up to constant, the Ruelle-Sullivan current associated with the holonomy invariant transverse measure Λ. To end this section, let us state the positivity result obtained in the case of foliations by four dimensional spin manifolds. Let D be the Dirac operator along the leaves and let γ be the Z2 -grading. The following proposition is a corollary of Proposition 9 and can be understood as furnishing a new inequality on the manifold M for any given measured foliation.


31

Proposition 10. For any hermitian vector bundle E over M and any hermitian connection ∇ on E, we have | < c1 (E)2 /2 − c2 (E), [CΛ ] > | ≤ YMΛ (∇) where c1 (E) and c2 (E) are the Chern classes of E. Proof. If we compute the Chern character of E and integrate it against the Ruelle-Sullivan current then we obtain the pairing of our Hochschild cocycle (which is cyclic here) with the K-theory of M by Theorem 7. Now Proposition 9 enables to conclude. Appendix A. Singular numbers We gather in this appendix some general facts about Dixmier traces associated with type II von Neumann algebras. We shall denote by M a von Neumann algebra acting on a Hilbert space H, and we shall assume that there exists a positive normal semi-finite faithful trace τ on M. A.1. τ -measurable operators. A densely defined closed operator T acting on H is said to be τ -measurable if it is affiliated with M and if there exists, for each ǫ > 0, a projection E in M such that E(H) ⊂ Dom(T ) and τ (1 − E) ≤ ǫ. Let T = U |T | be the polar decomposition of the densely defined closed operator T , and denote by Z +∞ |T | = λdEλ 0

the spectral decomposition of its module. Then, the operator T is τ -measurable if and only if both U and the Eλ′ s (λ ∈ R∗+ ) belong to M, and τ (1 − Eλ ) < +∞ for λ large enough. Let us also recall that the set of all τ -measurable operators is a ∗-algebra with respect to the strong sum, the strong product, and the adjoint of (densely defined) closed operators (cf. [12]).

A.2. τ -singular numbers. For any t > 0, the tth singular number (s-number) µt (T ) of a τ -measurable operator T is defined by : µt (T ) = Inf{kT Ek, E = E 2 = E ∗ ∈ M and τ (1 − E) ≤ t}. Thanks to the τ -measurability of T , we have : 0 ≤ µt (T ) = µt (|T |) < +∞ for any t > 0. There are several equivalent definitions of the singular numbers. For instance, we have (cf [26]): µt (T ) = Inf{λ ≥ 0, τ (1 − Eλ ) ≤ t}, R +∞ where |T | = 0 λdEλ still denotes the spectral decomposition of |T |, a fact which shows that the function t → µt (T ) is nothing but the non-increasing rearrangement of |T | as a positive measurable function on the measure space (sp(|T |) \ {0}, m). Here, sp(|T |) denotes the spectrum of |T | and m the spectral measure defined by m(B) = τ (1B (|T |)), (B Borel subset of sp(|T |) \ {0}). Note also that we have for any t > 0: µt (T ) = dist(T, Rt ), where Rt is the set of all τ -measurable operators S such that τ (supp(|S|)) ≤ t. This equality shows that the s-numbers may be viewed as a natural extension of the classical approximation numbers. For a detailed study of the generalized s-numbers, we refer to [26], where several spectral inequalities are proved. Let us just mention here the most useful of them, for the convenience of the reader: Lemma 7. (i) For any τ -measurable operator T , the function t → µt (T ) is non increasing and right continuous. Moreover, µt (T ) → kT k when t → 0; (ii) For any τ -measurable operator T , any complex number λ, and any t > 0, we have: µt (T ) = µt (|T |) = µt (T ∗ ) and µt (λT ) = |λ|µt (T )

32


(iii) For any τ -measurable operator T and non decreasing right-continuous function f on [0, +∞) such that f (0) ≥ 0, we have: µt (f (|T |)) = f (µt (T )), ∀t > 0; (iv) For any pair of τ -measurable operators T, S and for any t, s > 0, we have: µt+s (T + S) ≤ µt (T ) + µs (S) and µt+s (T S) ≤ µt (T )µs (S); (v) For any τ -measurable operator T , any pair of operators A, B ∈ M and any t > 0, we have : µt (AT B) ≤ kAkµt (T )kBk; (vi) For any pair of τ -measurable operators T, S satifying T ≤ S, we have: ∀t > 0, µt (T ) ≤ µt (S).

A.3. Non commutative integration theory. Let T be a τ -measurable operator. For any continuous increasing function f on [0, +∞) with f (0) = 0, we have: Z ∞ τ (f (|T |)) = f (µt (T ))dt. 0

[26][Corollary 2.8, page 278]. This basic relation explains why the s-numbers are of interest in the study of non commutative Banach spaces of functions such as Lp (M, τ ), p ≥ 1. In particular, we have T ∈ Lp (M, τ ) ⇔ [t → µt (T )] ∈ Lp ([0, ∞)) and

Z kT kp = (

∞

µt (T )p dt)1/p .

0

Most of the known s-numbers inequalities are based on the properties of the following function Z t µs (T )ds, s > 0. σt (T ) = 0

The following lemma gives three equivalent expressions of σt (T ).

Lemma 8. Let T be a τ -measurable operator. For any t > 0 we have σt (T ) = inf {kT1k1 + tkT2 k∞ , T = T1 + T2 , T1 ∈ L1 (M, τ ), T2 ∈ M}; and if M has no minimal projections, then we have: σt (T ) = sup{τ (E|T |E), E ∈ M, E 2 = E ∗ = E, τ (E) ≤ t}. Proof. The first interpolation formula is proved in [26], page 289 and the third equality also goes back to [26]. Proposition 11. If T1 , T2 are two positive τ -measurable operators then for (t1 , t2 ) ∈ R∗+ , we have σt1 (T1 ) + σt2 (T2 ) ≤ σt1 +t2 (T1 + T2 ) and σt (T1 + T2 ) ≤ σt (T1 ) + σt (T2 ). ˜ ˜ = M ⊗ L∞ ([0, 1], dt) and using the simple fact that µM (T ) = µM Proof. By imbeding M in M t t (T ⊗ id) we can assume that M has no minimal projections. If E1 , E2 are two projections in M such that τ (E1 ) = t1 and τ (E2 ) = t2 then the projection E = E1 ∨ E2 belongs to M and satisfies τ (E) ≤ t1 + t2 . We thus have

τ ((T1 + T2 )E) = τ (T1 E) + τ (T2 E) ≥ τ (T1 E1 ) + τ (T2 E2 ), thus lemma 8 gives the first inequality. The second one follows similarily from lemma 7 (iii), see for instance [26][Theorem 4.4 (ii)].


33

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43]

M. F. Atiyah, Elliptic operators, discrete groups and von Neumann algebras., Ast´ erisque 32/33, SMF, 1976 (1989), 79–92. M.F. Atiyah and I. M Singer, The index of elliptic operators, I, Anna. Math. 87 (1968), 484–530. M. Benameur, Foliated group actions and cyclic cohomology, preprint. M. Benameur and V. Nistor, Homology of complete symbols and non Commutative geometry, ”Collected papers on Quantization of Singular Symplectic Quotients”, Progress in Math. 198. M. Benameur and V. Nistor, Homology of algebras of families of pseudodifferential operators, to appear in J. of Funct. Analysis. M. Benameur, T. Fack and V. Nistor, The local measured index theorem for foliations, work in progress. N. Berline, E. Getzler and M. Vergne, Heat kernels and Dirac operators, Springer-Verlag 1991. Bourbaki, Integration Fasc. 25, livre 6, chapitre 6. M. Breuer, Fredholm theories in von Neumann algebras. I, Math. Ann. 178 (1968), 243-254. M. Breuer, Fredholm theories in von Neumann algebras. II, Math. Ann. 180 (1969), 313-325. A. Carey and J. Phillips, Unbounded Fredholm modules and spectral flow, Can. J. Math. Vol. 50, (1998), 673-718. L.A. Coburn, R.D. Moyer and I.M. Singer, C ∗ -algebras of almost periodic pseudo-differential operators, Acta Math. 130 (1973), 279-307. A. Connes. Sur la th´ eorie non commutative de l’integration. Alg` ebres d’op´ erateurs, LNM 725, Springer Verlag, 19–143, 1979. A. Connes. The action functional in noncommutative geometry. Comm. Math. Phys. 117, (1988), 673–683. A. Connes. Noncommutative Geometry. Academic Press, New York - London, 1994. A. Connes. Cours au Coll` ege de France, 1990. A. Connes, Non-commutative differential geometry, Publ. IHES 62, 257-360 (1986). A. Connes and J. Cuntz, Quasi homomorphismes, cohomologie cyclique et positivit´ e, Commun. Math. Phys. 114, 515-526 (1988). A. Connes and H. Moscovici, The local index formula in nnoncommutative geometry, Geom Funct. An. Vol 5, No 2 (1995). A. Connes and H. Moscovici, Hopf algebras, cyclic cohomology and the transverse index theorem, preprint. J. Cuntz, Bivariante K-Theorie fr lokalkonvexe Algebren und der Chern-Connes-Charakter. (German) [Bivariant K-theory for locally convex algebras and the Chern-Connes character], Doc. Math. 2 (1997), 139–182. J. Dixmier, Les alg` ebres d’op´ erateurs dans l’espace hilbertien (Alg` ebres de von Neumann), Gauthier-Villars, Paris, 1957. J. Dixmier, Existence de traces non normales, CRAS Paris 262 (1966), A1107-A1108. K. J. Dykema, N. J. Kalton, Spectral characterization of sums of commutators II., J. Reine Angew. Math. 504 (1998), 127–137. T. Fack, Sur la notion de valeur caract´ eristique. JOT 7, (1982), 307-333. T. Fack and H. Kosaki, Generalized s-numbers of measured operators, Pac. J. Math. 123 (1986), 269-300. D. Guido and T. Isola, Singular traces on semifinite von Neumann algebras, J. funct. Anal. 134 (1995), 451-485. D. Guido and T. Isola, Noncommutative Riemann integration and Novikov-Shubin invariants for open manifolds, J. funct. Anal. 176 (2000), 115-152. R. Haag, Local quantum physics Fields, particles, algebras. Second edition. Texts and Monographs in Physics. SpringerVerlag, Berlin, 1996. C. Kassel, Le r´ esidu non commutatif, S´ eminaire Bourbaki, 1988-89, No 708. J-L. Loday, Cyclic homology, Springer-Verlag, Berlin-Heidelberg-New York, 1992. MacLane, Homology, Springer-Verlag, Berlin-Heidelberg-New York, 1995. J. Milnor, Introduction to algebraic K-theory, Ann. of Math. Stud., 72, Princeton Univ. Press (1971). C.C Moore and C. Schochet, Global analysis on Foliated spaces, Springer-Verlag 1988. V. Nistor and A. Weinstein, and Ping Xu, Pseudodifferential operators on groupoids, Penn State Preprint No. 202 (1997). J-F. Plante, Foliations with measure preserving holonomy, Ann. Math. 102, 327-361 (1975). R. Prinzis, Traces r´ esiduelles et asymptotique du spectre d’op´ erateurs pseudo-diff´ erentiels., Th` ese de Doctorat de l’Universit´ e de Lyon, 1995. D. Ruelle and D. Sullivan Currents, flows and diffeomorphisms, Topology 14, 319-327 (1975). M. A. Shubin,Pseudodifferential almost periodic operators and von Neumann algebras, Trnans. Moscow Math. Soc. 1 (1979) 103-166. M. A. Shubin,The spectral theory and the index of elliptic operators with almost periodic operators, Russian Math. Survey 34:2 (1979) 109-157. B. Tsygan,Homology of matrix Lie algebras over rings and Hochschild homology, Uspekhi Math. Nauk., 38 (1983), 217-218. J. M. Gracia-Bondia, J. C. Varilly and H. Figueroa, Elements of Noncommutative Geometry, Birkhauser advanced texts, (2001). D. V. Widder, The Laplace transform, Princeton Univ. Press (1946).

34


Inst. Desargues, Lyon, France E-mail address: [email protected] Inst. Desargues, Lyon, France E-mail address: [email protected]

TYPE II NON COMMUTATIVE GEOMETRY. I. DIXMIER TRACE IN

TYPE II NON COMMUTATIVE GEOMETRY. I. DIXMIER TRACE IN

Suggest Documents

Manifest Supersymmetry in Non-Commutative Geometry

Gravity in Non-Commutative Geometry - Project Euclid

SO (10) Unification in Non-Commutative Geometry

Differential Geometry in Non-Commutative Worlds

Non-Commutative Geometry, Non-Associative Geometry and the ...

INDEX THEORY AND NON-COMMUTATIVE GEOMETRY I. HIGHER ...

Non-commutative Korovkin-type theorems

Supersymmetric quantum theory and non-commutative geometry

Non-commutative symplectic NQ-geometry and ...

Wormhole inspired by non-commutative geometry - Core

(De) Constructing Dimensions and Non-commutative Geometry

Non-Commutative Geometry, Categories and Quantum Physics

(De) Constructing Dimensions and Non-commutative Geometry

Gauge theories and non-commutative geometry

Non-Commutative Geometry, Categories and Quantum Physics

Non-commutative geometry and matrix models

Super-connections and non-commutative geometry

Supersymmetric quantum theory, non-commutative geometry, and ...

Non-commutative differential geometry - Alain Connes

NON COMMUTATIVE FINITE DIMENSIONAL MANIFOLDS II

Yukawa Couplings in F-theory and Non-Commutative Geometry

The Chern-Simons Action in Non-Commutative Geometry

Non-commutative desingularization of determinantal varieties, I

Bi-Local Higgs-Like Fields Based on Non-Commutative Geometry