The Kadison-Kaplansky conjecture for word-hyperbolic groups · The Kadison-Kaplansky conjecture for word-hyperbolic groups Michael Puschnigg Institut de Math´ematiques de Luminy,

Digital Object Identifier (DOI) 10.1007/s002220200216Invent. math. 149, 153–194 (2002)

The Kadison-Kaplansky conjecture forword-hyperbolic groups

Michael Puschnigg

Institut de Mathematiques de Luminy, CNRS, UPR 9016, Case 907, 163 Avenue de Luminy,13288 Marseille Cedex 9, France

Oblatum 3-V-2001 & 28-XI-2001Published online: 15 April 2002 – Springer-Verlag 2002

Introduction

In this paper we prove the Kadison-Kaplansky idempotent conjecture fortorsion-free word-hyperbolic groups. The conjecture asserts that the follow-ing equivalent statements hold for a torsion-free discrete group Γ:

• The reduced group C∗-algebra C∗r (Γ) contains no idempotents except 0and 1.

• The spectrum of every element of the reduced group C∗-algebra isconnected.

• The canonical trace on C∗r (Γ) takes integer values on idempotents.

The last assertion can be viewed as a statement about the pairing betweenthe K-theory and the (local) cyclic cohomology of the group C∗-algebra.It is in this setting that we will prove the conjecture. Our proof is basedon a partial analysis of the assembly maps in K-theory and local cyclichomology. We compare these assembly maps by means of an equivariantbivariant Chern-Connes character.

Before going into details, we recall some previous work on the conjec-ture. The first progress was achieved by Pimsner and Voiculescu [PV] whoproved the Kadison-Kaplansky conjecture for free groups as a consequenceof their computation of the K-theory of the group C∗-algebra. Subsequentlyit was realized that more generally the Kadison-Kaplansky conjecture wasa consequence of the Baum-Connes conjecture which gives a geometricdescription of K∗(C∗r (Γ)) for any torsion-free discrete group. In fact theBaum-Connes conjecture states that the K-theoretic assembly map

µ : Ktop∗ (BΓ) −→ K∗(C∗r (Γ))

is an isomorphism [BCH]. In particular, the Kadison-Kaplansky conjectureholds for all torsion-free groups, for which the Baum-Connes conjecture is

154 M. Puschnigg

known to be true [HK], [La]. Recently Mineyev-Yu [MY] and Lafforgue(unpublished) proved the Baum-Connes conjecture for word-hyperbolicgroups. Their work therefore also yields a proof of the Kadison-Kaplanskyconjecture for torsion-free word-hyperbolic groups.

A different approach to the idempotent conjecture, based on cyclic coho-mology, was initiated by Connes in [Co]. Connes realized that to prove theKadison-Kaplansky conjecture for free groups one could introduce a Fred-holm module, implicit in the work of Pimsner and Voiculescu. Connesshowed that the Chern-character of this Fredholm module, defined throughhis theory of cyclic cohomology, coincided with the canonical trace ofC∗r (Γ). This immediately implied the conjecture.

The use of cyclic homology as a tool for studying the assembly map inK-theory goes back to Connes and Moscovici. In [CM] they established forany torsion-free discrete group Γ a natural commutative diagram

µ : Ktop∗ (BΓ) −−−→ K∗(a(Γ))

ch

ch

µcoh : H∗(Γ, HP∗(C)) −−−→ HP∗(a(Γ))

where µ is the K-theoretic assembly map and a(Γ) denotes any Banachalgebra completion of the complex group ring CΓ inside the reduced groupC∗-algebra C∗r (Γ). Connes and Moscovici use this to deduce the Novikovconjecture for word-hyperbolic groups from the injectivity of the homolog-ical assembly map µcoh .

The arguments of Connes and Moscovici apply immediately so that thediagram above remains valid if periodic cyclic homology is replaced bylocal cyclic homology [Pu1]. Our whole effort concentrates on studying thelower line

µcoh : H∗(Γ, HCloc∗ (C)

) −→ HCloc∗ (a(Γ))

of the diagram for appropriate a(Γ). Properties of the assembly map are de-duced from a detailed explicit calculation of HCloc∗ (a(Γ)) using homologicalmethods.

We work with local cyclic cohomology for two reasons. On the onehand bivariant local cyclic cohomology of Banach algebras [Pu1] is knownto be a bifunctor which is stable, satisfies excision [Pu2], and is invariantnot only under smooth but even under continuous homotopies. The lastproperties make local cyclic cohomology much better behaved on the cat-egory of Banach-algebras than other cyclic cohomology theories. On theother hand, it is also explicitely computable in a certain sense. In fact,if a Banach algebra A satisfies the metric approximation property andis the topological direct limit of a countable directed family (An)n∈N ofBanach algebras with nuclear transition maps, then the local cyclic ho-mology of A is given as the direct limit of the entire cyclic homologygroups HCε∗(An) [Co2] of the algebras An . At first glance one might seem

The Kadison-Kaplansky conjecture for word-hyperbolic groups 155

to have replaced one intractable problem by another, since entire cyclicgroups are notoriously difficult to calculate. However the direct limit onlyrequires information on the transition maps which in contrast are amenableto study.

In particular we can determine HCloc∗ (1(Γ)) by this strategy because1(Γ) is the topological direct limit of the Banach algebras of summablefunctions of exponential decay (with respect to a word metric) on Γ [Bo].

The cyclic homology of group rings has been determined by Burghe-lea [Bu] and Nistor [Ni], and is closely related to group homology. By theirworks, one knows that the Hochschild complex ofCΓ can be identified withthe bar complex of Γ twisted by the adjoint representation. Moreover theadjoint representation decomposes canonically as a direct sum of a homo-geneous and an inhomogeneous part, corresponding to the unit respectivelyto the nontrivial group elements. There is a corresponding canonical decom-position of the various cyclic complexes of good completions of CΓ. Onlythe homogeneous parts are required for the proof of the Kadison-Kaplanskyconjecture.

In order to construct a contracting homotopy of the bar complex of Γ inlarge dimensions, we compare the bar resolution of the constant Γ-modulewith the resolution coming from a Rips complex [Gr]. As a consequence wesee that algebraic differential forms with degree higher than the dimensionof the Rips complex do not contribute to HP∗(CΓ). The continuity proper-ties of the contracting homotopy allow to establish the same assertion forHCloc∗ (1(Γ)). After this it is not difficult to identify the homogeneous partHCloc∗ (1(Γ))hom with the group homology H∗(Γ, HCloc∗ (C)). We then usean argument of Connes and Moscovici [CM] to complete the proof of thefirst main result of the paper: the homological assembly map

µcoh : H∗(Γ, HCloc∗ (C)

) −→ HCloc∗ (a(Γ))

is an isomorphism onto the homogeneous part HCloc∗ (a(Γ))hom of the localcyclic homology when a(Γ) is a good completion ofCΓ (Γ torsion-free andword-hyperbolic).

After this we use the diagram of Connes and Moscovici above to comparethe assembly maps in K-theory and local cyclic homology. In order to carrythis out we introduce equivariant bivariant Chern-Connes characters. Theseare natural transformations

chΓbiv : KKΓ(−,−) −→ HCloc

0 (−r Γ,−r Γ)

and

cha(Γ)biv : KKΓ(−,−) −→ HCloc

0 (a(Γ,−), a(Γ,−))

from Kasparov’s equivariant KK-theory [Ka1], [Ka2] to bivariant localcyclic cohomology which are multiplicative and compatible withdescent [Ka2], with ordinary Chern characters and with the canonicaldecomposition into homogeneous and inhomogeneous parts. (Here a(Γ)

156 M. Puschnigg

denotes any good completion of CΓ.) The existence of these Chern-Connescharacters is an immediate consequence (see [Cu1]) of the characterizationof equivariant KK-theory as the universal stable and split exact homotopyfunctor on the category of separable Γ-C∗-algebras [Th].

The second main result of this paper states that the decomposition ofK∗(a(Γ)) into the direct sum of the image and the cokernel of the assemblymap, given by a γ -element, is compatible under the Chern character withthe canonical decomposition of HCloc∗ (a(Γ)). More precisely, one has

Theorem 0.1 Let Γ be a torsion-free discrete group. Suppose that a(Γ) isa good completion of CΓ such that the homological assembly map definesan isomorphism µcoh : H∗(Γ, HCloc∗ (C))

−→ HCloc∗ (a(Γ))hom. Let γ ∈KKΓ(C,C) be a γ -element for Γ. Then the equivariant Chern-Connescharacter

cha(Γ)biv (γ) ∈ HCloc

∗ (a(Γ), a(Γ))

acts on the local cyclic homology HCloc∗ (a(Γ)) as the canonical projectiononto the homogeneous part HCloc∗ (a(Γ))hom.

In particular, such good completions contain no idempotents: the canon-ical trace τ is a homogeneous cocycle and therefore vanishes onch((1− γ) K∗(a(Γ))), since from the theorem it is inhomogeneous. Hence

τ(K∗(a(Γ))) = τ(ch(γ · K∗(a(Γ)))) ⊂ Zby the L2-index theorem of Atiyah and Singer [At], [Si].

In particular if Γ is torsion-free and word-hyperbolic such good a(Γ)exist. Moreover a(Γ) can be chosen to be closed under the holomorphicfunctional calculus in C∗r (Γ) [Jol]. Hence the Kadison-Kaplansky conjectureis true for all such Γ.

I am indebted to A. Connes and G. Kasparov for suggesting that I adaptthe methods of [Pu3] to word-hyperbolic groups. It is a pleasure for meto thank G. Skandalis and V. Lafforgue for constructive comments on thecontent of this paper and for bringing the work of Mineyev-Yu and ofLafforgue to my attention. I heartly thank A. Wassermann for numeroussuggestions improving clarity and style of the exposition.

Contents

1 Hyperbolic spaces and hyperbolic groups . . . . . . . . . . . . . . . . . . . . . 1572 Controlled resolutions of hyperbolic groups . . . . . . . . . . . . . . . . . . . . 1593 Local cyclic homology of group Banach algebras . . . . . . . . . . . . . . . . . 1664 Auxiliary results about crossed products and their local cyclic homology . . . . . 1775 Equivariant Chern-Connes characters and the Kadison-Kaplansky conjecture . . . 184


1. Hyperbolic spaces and hyperbolic groups

We recall a few basic notions of Gromov’s theory of hyperbolic metricspaces and groups.

Definition 1.1 (Gromov) [Gr] A geodesic metric space X is called δ-hyperbolic if any four points x, y, u, v ∈ X satisfy

d(x, y)+ d(u, v) ≤ maxd(x, u) + d(y, v), d(x, v) + d(y, u) + 2δ

An important property of hyperbolic geodesic metric spaces, reminiscentof nonpositively curved manifolds is

Proposition 1.2 (Gromov) [Gr] The distance function on a δ-hyperbolicgeodesic metric space is 6δ-convex: if z0 := (1 − t)x + ty and z1 :=(1− t)u + tv are points on geodesic segments [x, y] and [u, v] dividing thesegments in the ratio t : (1− t), then

d(z0, z1) ≤ (1− t)d(x, u) + td(y, v)+ 6δ

The proof can be found in [Gr], (7.4).

Definition 1.3 A straight k-polyhedron 〈x0, . . ., xk〉 with vertices x0, . . ., xkin a geodesic metric space X is a subset of X which is the union 〈x0, . . . , xk〉= ⋃

z∈〈x1,... ,xk〉[x0, z] of geodesic segments with origin x0 and endpoint z ∈

〈x1, . . . , xk〉. A straight zero polyhedron with vertex x0 is just the one pointset x0 ⊂ X.

Corollary 1.4 Let 〈x0, . . . , xk〉 be a straight k-polyhedron in a δ-hyperbolicgeodesic metric space X and let z′, z′′ ∈ X. Then

supz∈〈x0,... ,xk〉

(d(z, z′)+ d(z, z′′)) ≤ max0≤i≤k

(d(xi, z′)+ d(xi, z′′))+ 12kδ

The proof is straightforward from the 6δ-convexity of the distance function.

Definition 1.5 (Rips-complex) [Gr]

a) For a nonempty set X let ∆(X) be the simplicial set with n-simplices∆(X)n := Xn+1 and with face and degeneracy maps ∂i(x0, . . . , xn) :=(x0, . . . , xi , . . . , xn) and si(x0, . . . , xn) := (x0, . . . , xi, xi, . . . , xn).The simplicial set ∆(X) is contractible.

b) Let X be a metric space and let R > 0. The Rips complex PR(X) is thesimplicial subset of ∆(X) with n-simplices

PR(X)n := (x0, . . . , xn) ∈ ∆(X)n|d(xi, x j) < R, ∀i, jThe construction of the simplicial set ∆(X) is functorial in X. In particu-

lar any group acting on X induces a simplicial action on ∆(X). Similarly

158 M. Puschnigg

any isometric action of a group on a metric space X induces a simplicialaction of the group on the Rips complexes PR(X).

The Rips complex of a hyperbolic metric space turns out to be usefulbecause of

Proposition 1.6 (Rips) [Gr] Let X be a δ-hyperbolic geodesic metric spaceand let Y be an ε-dense subset of X. Then PR(Y ) is contractible for R >4δ + 6ε.

The proof can be found in [Gr], (1.7). It will also be recalled for the Ripscomplex of a hyperbolic group in 2.4.

Let now Γ be a finitely generated group and let S be a finite symmetricset of generators of Γ. Denote by |g| := lS(g) the word length with respectto S of an element g ∈ Γ and let dS be the left invariant word metricdS(g, g′) := lS(g−1g′).

Definition 1.7 (Gromov) [Gr] A finitely generated group (Γ, S) is calledword-hyperbolic if its Cayley graph with respect to S is δ-hyperbolic forsome δ ≥ 0.

If Γ is word-hyperbolic with respect to one set of generators it is so for anyother set of generators so that hyperbolicity is an intrinsic notion of a group.

A finitely generated group (Γ, S) acts simplicially and properly on theRips complex PR(Γ, dS) for any R > 0. If Γ is torsion-free then it acts freelyand the geometric realization of PR(Γ)/Γ is a finite simplicial complex.

The Rips complex of a finitely generated group has no particularlyinteresting properties. On the contrary, if Γ is a word-hyperbolic groupin the sense of Gromov, then the Rips complex PR(Γ) is contractible forR 0 (1.6) and is therefore a universal proper Γ-space. This impliesfor example that a torsion-free, word-hyperbolic group possesses a finitesimplicial complex as classifying space.

As a group is 1-dense in any of its Cayley graphs, (1.6) shows thatthe Rips complex PR(Γ) of a δ-hyperbolic group (Γ, S) is contractible forR > 4δ + 6. We describe now a simplicial homotopy operator that will beused in (2.5) to construct an explicit contracting homotopy of the singularchain complex of PR(Γ).

Definition and Lemma 1.8 [Gr]

a) Let (Γ, S) be a finitely generated group and let R > 0 be an even integer.Let σ : Γ → Γ be any map such that σ(g) = e if |g| ≤ R

2 and such thatotherwise σ(g) lies on a geodesic segment [g, e] joining g and e in theCayley graph at distance R

2 from g. Note that σ is by no means unique.b) Let h : ∆(Γ)∗ → ∆(Γ)∗+1 be defined by

h(g0, . . . , gi, . . . , gn) := (−1)i(g0, . . . , σ(gi), gi, . . . , gn)

where gi is characterized by the conditions |gi |=maxj|gj | and |gi′ |< |gi|,

i ′ < i.


c) Suppose that (Γ, S) is δ-hyperbolic and let R > 4δ + 6 be an eveninteger. Then the operator h of b) preserves the Rips complex PR(Γ), i.e.it defines a map h : PR(Γ)∗ → PR(Γ)∗+1.

We end this section with the

Remark 1.9 Let (Γ, S) be a finitely generated group and let σ : Γ → Γ bea map as constructed in (1.8). Denote by m the action of Γ by left translationand let (g0, . . . , gk) ∈ Γk+1. Then the set(

m(g0)σj0m(g−1

0 g1). . . σ jk−1m

(g−1

k−1gk))

(e)∣∣ j0 ≥ 0, . . . jk−1 ≥ 0

is contained in a straight k-polyhedron 〈g0, . . . , gk〉 in the Cayley graph of(Γ, S) with vertices g0, . . . , gk .

2. Controlled resolutions of hyperbolic groups

Bar resolution and Rips resolutions

Let Γ be an abstract group. We recall two well known projective resolutionsof the constant Γ-module C. The first is the bar resolution, which providesa universal resolution whereas the second resolution, obtained from the Ripscomplex, is defined for word-hyperbolic groups in the sense of Gromov, andreflects the geometry of the given group.

Definition and Lemma 2.1 Let Γ be a discrete group and let ∆(Γ) be thesimplicial set introduced in (1.5).

a) The bar resolution C∗(Γ) of the constant Γ-module C is the chain com-plex of complex vector spaces associated to the simplicial Γ-set ∆(Γ).Thus Cn(Γ) is the complex vector space generated by the set of n-simplices ∆(Γ)n and the differential of C∗(Γ) is given by the alternatingsum of the face maps of ∆(Γ).

b) The reduced bar resolution C ∗(Γ) is the quotient of the full bar resolutionby the subcomplex spanned by the degenerate simplices of ∆(Γ).

c) Both resolutions are augmented by the Γ-map C0(Γ) → C sendinga simplex g ∈ Γ to 1 and define resolutions of the constant Γ-module Cby free Γ-modules.

d) A natural Γ-basis of the (reduced) bar resolution is given by the simplicesof the form (e, g1, . . . , gn). A contracting linear homotopy is given bys(g0, . . . , gn) := (−1)n+1(g0, . . . , gn, e).

e) The antisymmetrization operator π defined on simplices byπ(g0, . . . , gn) := 1

(n+1)!∑

σ∈Σn+1

(−1)sig(σ)(gσ(0), . . . , gσ(n)) is a map of

(reduced) bar resolutions. It equals the identity in degree zero.

Definition and Lemma 2.2 Let Γ be a finitely generated group with finitesymmetric set of generators S and let R > 0.

160 M. Puschnigg

a) Denote by C R∗ (Γ) the chain complex of complex vector spaces associatedto the simplicial Γ-set PR(Γ). It is a complex of free Γ-modules anda subcomplex of the full bar resolution C∗(Γ).

b) Denote by CR∗ (Γ) the quotient chain complex of C R∗ (Γ) by the subcomplex

generated by degenerate simplices. It is a complex of free Γ-modules anda subcomplex of the reduced bar resolution C∗(Γ).

c) The complexes C R∗ (Γ) and CR∗ (Γ) are stable under antisymmetrization.

Lemma 2.3 Let (Γ, S) be a finitely generated group and let h : ∆(Γ)∗ →∆(Γ)∗+1 be a homotopy operator as constructed in (1.8). Denote the cor-responding linear operator on C∗(Γ) by the same letter.

a) The operator h maps degenerate simplices to degenerate simplices andthe operators h and ϕ := Id− (h∂+ ∂h) descend therefore to operatorson the reduced bar resolution C∗(Γ).

b) Put

χ :=∞∑

n=0

h ϕn : C∗(Γ)→ C∗+1(Γ)

and letϕ∞ := Id − (χ∂ + ∂χ) = lim

n→∞ϕn

Then ϕ∞ defines a deformation retraction of C∗(Γ) onto the constant Γ-moduleC and χ is a contracting homotopy of the reduced bar resolution.

c) The homotopy operator χ satisfies χ2 = 0.d) For a reduced bar simplex (g0, . . . , gn) one has

χ(g0, . . . , gn) =∑

±(g′0, . . . , g′n+1)

where the simplices (g′0, . . . , g′n+1) satisfy maxj|g′j | ≤ max

i|gi | and the

number of summands is less than or equal to C(∑n

i=0 |gi |) for a constantC depending only on (Γ, S) and the choice of h.

Proof: Assertion a) is clear from the definition of h. For c) note that byconstruction the image of h2 consists of degenerate simplices so that h2 = 0on C∗(Γ). Thus Im h is stable under ϕ as ϕ h = (Id− h ∂− ∂ h) h =h (Id − ∂ h). This implies already that χ2 = 0.

Let (g0, . . . , gn) be a simplex in CR∗ (Γ). A straightforward calculation

shows

ϕ(g0, . . . , gi, . . . , gn)

= (g0, . . . , σ(gi), . . . , gn)− (−1)ih(g0, . . . , gi, . . . , gn)

if gi−1 = σ(gi), i.e. if (g0, . . . , gn) is not in the image of h and

ϕ(h(g0, . . . , gi, . . . , gn)) = (−1)i+1h(g0, . . . , σ(gi), gi, . . . , gn)


otherwise. These formulas show that ϕ∞ is zero on C∗(Γ), ∗ > 0 andprojects C0(Γ) onto C(e). As it is by definition chain homotopic to the iden-tity it is a deformation retraction as claimed in b). The formulas aboveshow moreover that h ϕk(g0, . . . , gn) = 0 if (k − l) R

2 >∑n

i=0 |gi |where l = #i|gi = e. (Here R is the constant chosen in (1.8).) More-over ϕk(g0, . . . , gn) consists of a single simplex modulo Im h so that±h ϕk(g0, . . . , gn) is actually given by a single simplex. This showsthat χ(g0, . . . , gn) =∑∞

k=0 h ϕk(g0, . . . , gn) is of the form claimed in d).

If Γ is word-hyperbolic the simplicial homotopy operator h above isknown to preserve the Rips complex PR(Γ) for large R (1.8). It follows thatthe contracting homotopy χ of the reduced bar complex of (2.3) yields alsoa contracting homotopy of the reduced cellular chain complex of the Ripscomplex.

Corollary 2.4 [Gr] Let Γ be a word-hyperbolic group in the sense ofGromov and let S a finite symmetric set of generators of Γ. Let R 0 belarge and let h and χ be homotopy operators as in (1.8) and (2.3). Thenχ defines a contracting homotopy of the reduced cellular chain complex

CR∗ (Γ) of the Rips complex PR(Γ) onto its base point. In particular, C

R∗ (Γ)

defines a resolution of the trivial Γ-module C by free Γ-modules. Any suchresolution will be called a Rips resolution.

Lemma 2.5 Let Γ be a word-hyperbolic group with finite symmetric set ofgenerators S. Let R 0 be large enough that the Rips complex PR(Γ) iscontractible and let χ be a contracting homotopy of it as in (2.3). Fix aninteger d > #g ∈ Γ, |g| < R.a) There exists a Γ-equivariant chain map Φ′ : C∗(Γ)→ C

R∗ (Γ) ⊂ C∗(Γ)

which equals the identity in degree zero and is given on simplices by theformula

Φ′(g0, . . . , gn) :=(m(g0) χ m

(g−1

0 g1) . . . χ m

(g−1

n−1gn))

(e)

Here m(g) denotes the left translation of g ∈ Γ on C∗(Γ).

b) Let Φ : C∗(Γ) → CR∗ (Γ) ⊂ C∗(Γ) be the composition of Φ′ and the

antisymmetrization operator: Φ := π Φ′. Then Φ is a Γ-equivariantchain map which equals the identity in degree zero and vanishes indegrees larger or equal to d.

Proof: The reduced bar resolution C∗(Γ) of the constant Γ-module C con-sists of free Γ-modules with a natural basis described in (2.1) d). If Γ isa word-hyperbolic group there exist on the other hand the Rips resolutionsC

R∗ (Γ) of the constant Γ-module C which coincide with the reduced bar

resolution in degree zero and possess a contracting C-linear homotopy χ.A well known argument from homological algebra allows to extend the

162 M. Puschnigg

identity in degree zero inductively to a chain map from the reduced barresolution to the Rips resolution. Because the homotopy operator χ satisfiesχ2 = 0 (2.3)c) this chain map of complexes of Γ-modules can be given bythe explicit formula in a). As the antisymmetrization operator π commuteswith the group action and equals the identity in degree zero it remains onlyto verify that π annihilates simplices in PR(Γ) of degree larger or equal to d.This is clear however as the factors of such a simplex cannot be pairwisedifferent by the choice of d.

Operators on twisted bar resolutions

Definition 2.6 Let Γ be a finitely generated group.

a) Let Ad(Γ) be the Γ-space with underlying set Γ equipped with the adjointaction. Denote by C∗(Γ)⊗Ad(Γ) the Γ-chain complex of complex vectorspaces associated to the simplicial Γ-set ∆(Γ)×Ad(Γ) with the diagonalΓ-action.

b) Let B′ : C∗(Γ) ⊗ Ad(Γ) → C∗+1(Γ) ⊗ Ad(Γ) be the Γ-equivariantlinear operator given on simplices by the formula

B′(g0, . . . , gn, v) :=n∑

i=0

(−1)n(i+1)(gi, gi+1, . . . , gn, vg0, vg1, . . . , vgi, v)

c) Suppose that Γ is word-hyperbolic and let Φ be a Γ-equivariant chainmap as in (2.5). Let ∇′ : C∗(Γ) ⊗ Ad(Γ) → C∗+1(Γ) ⊗ Ad(Γ) be theΓ-equivariant linear operator given on simplices by

∇′(g0, . . . , gn, v) := (−1)nn∑

i=0

(Φ(g0, . . . , gi), gi, . . . , gn, v)

Note that the number of summands is bounded independently of n be-cause Φ vanishes in large degrees.

Lemma 2.7 The operator ∇′ of (2.6) is a Γ-equivariant contracting homo-topy of C∗(Γ)⊗ Ad(Γ) for ∗ ≥ d the constant introduced in (2.5).

Proof: According to (2.5) the Γ-equivariant chain map Id −Φ : C∗(Γ) →C∗(Γ) equals the identity in degrees larger or equal to d and vanishes indegree zero. As C∗(Γ) is a complex of free Γ-modules with canonical Γ-basis (2.1) and is linearly contractible (2.1)d) the map Id−Φ is canonicallynullhomotopic. The same holds for the operator (Id−Φ)⊗ Id on C∗(Γ)⊗Ad(Γ). The operator ∇′ is the corresponding homotopy operator.


Norm estimates

Now the continuity properties of the operators constructed so far will bestudied. We begin by introducing a family of norms on the bar resolution ofa finitely generated group.

Definition 2.8 Let (Γ, S) be a finitely generated group.

a) For a simplex (g0, . . . , gn, v) ∈ ∆(Γ)n × Ad(Γ) put

|(g0, . . . , gn, v)| := dS(g0, g1)+ . . .+ dS(gn−1, gn)+ dS(gn, vg0)

b) For a monotone increasing function f : R+ → R+ let ‖ − ‖ f be thelargest seminorm on C∗(Γ)⊗ Ad(Γ) satisfying

‖ (g0, . . . , gn, v) ‖ f ≤ f(|(g0, . . . , gn, v)|)c) The seminorms associated to the functions f(t) := λt, λ > 1 will be

denoted by ‖ − ‖λ.

The action of Γ on C∗(Γ) ⊗ Ad(Γ) is isometric with respect to the norms‖ ‖ f .

Proposition 2.9 Let (Γ, S) be a word-hyperbolic group. Let Φ be a chainmap as in (2.5) and let ∇′, B′ be the operators introduced in (2.6). Fixa number d such that the chain map Φ vanishes in degrees larger than d.Then there exist constants C0, C1 such that for every monotone increas-ing function f : R+ → R+ and every αn ∈ Cn(Γ) ⊗ Ad(Γ), n ∈ N, theestimates

‖ ∇′(αn) ‖ f ≤ C0 ‖ αn ‖ f ′

and

‖ ∇′ B′(αn) ‖ f ≤ C0(n + 1)1−d ‖ αn ‖ f ′

hold for f ′ the function f ′(t) := td f(t + C1).

Proof: The norms ‖ − ‖ f are weighted 1-norms so that it suffices to verifythe estimates on simplices (g0, . . . , gn, v) ∈ ∆(Γ)n × Ad(Γ). The explicitformulas in (2.3) and (2.5) show that

(Φ′(g0, . . . , gk), gk, . . . , gn, v) =∑

±(g′0, . . . , g′k, gk, . . . , gn, v)

is an alternating sum of at most

Ck · k! · (dS(g0, g1)+ . . .+ dS(gk−1, gk))k

simplices (g′0, . . . , g′k, gk, . . . , gn, v) of the following form:

• k ≤ d• (g′0, . . . , g′k) is a Rips simplex, i.e. dS(g′i, g′j) < R, ∀i, j

164 M. Puschnigg

• The factor g′k is of the form

g′k = m(g0)σj0m(g−1

0 g1). . . σ jk−1m

(g−1

k−1gk)(e)

and is therefore by (1.9) contained in a straight k-polyhedron with ver-tices (g0, . . . , gk) in the Cayley-graph of (Γ, S).

In particular

|(g′0, . . . , g′k, gk, . . . , gn, v)|

=k−1∑i=0

dS(g′i, g′i+1)+

n−1∑j=k

dS(gj, gj+1)+ dS(g′k, gk)+ dS(gn, vg′0)

≤n−1∑j=k

dS(gj, gj+1)+ dS(gn, vg′k)+ dS(g′k, gk)+ (k + 1)R

which can be estimated after (1.4) by

≤n−1∑j=k

dS(gj , gj+1)+ max0≤i≤k

(dS(gn, vgi)+ dS(gi, gk))+ (k + 1)R + 12kδ

≤ |(g0, . . . , gn, v)| + (k + 1)R + 12kδ

A similar estimate holds after antisymmetrization in the first k variables for(Φ(g0, . . . , gk), gk, . . . , gn, v) so that altogether

‖ (Φ(g0, . . . , gk), gk, . . . , gn, v) ‖ f ≤≤ C(k)(dS(g0, g1)+ . . .+ dS(gk−1, gk))

k f(|(g0, . . . , gn, v)| + C ′(k))

From this one obtains

‖∇′(g0, . . . , gn, v) ‖ f

≤ C ′(d)(dS(g0, g1)+ . . .+ dS(gn−1, gn))d f(|(g0, . . . , gn, v)| + C ′′(d))

which establishes the first inequality.Concerning the second note that

|(gi, gi+1, . . . , gn, vg0, vg1, . . . , vgi, v)| = |(g0, . . . , gn, v)|which implies

‖ ∇′ B′(g0, . . . , gn, v) ‖ f≤

≤ C ′ f(|(g0, . . . , gn, v)| + C ′′)n∑

i=0

(dS(gi, gi+1)′ + . . .+ dS(gi+d , gi+d+1)

′)d

where the indices i, . . . , i + d + 1 ∈ Z/(n + 1)Z are counted in cyclicorder and dS(gi, gi+1)

′ := dS(gi, gi+1) for 0 ≤ i < n and dS(gn, g0)′ :=

dS(gn, vg0).


If x0, . . . , xn are positive real numbers, then

n∑i=0

(xi + xi+1 + . . .+ xi+d)d ≤ C ′′(d) · (n + 1)1−d · (x0 + . . .+ xn)

d

where the indices are counted in cyclic order. To see this one can assume byhomogeneity that x0+. . .+xn = n+1. A simple analysis shows that the lefthand side attains its maximum under these conditions for x0 = . . . = xn = 1and in this case the claimed inequality is obvious. Applying it one finds

‖ ∇′ B′(g0, . . . , gn, v) ‖ f≤≤ C0(n + 1)1−d · |(g0, . . . , gn, v)|d · f(|(g0, . . . , gn, v)| + C1)

which proves the second inequality. The aim of this section is to establish

Corollary 2.10 Let (Γ, S) be a word-hyperbolic group and let ∇′, B′ bethe operators on C∗(Γ) ⊗ Ad(Γ) introduced in (2.6). Let λ0, λ1 be a pairof real numbers with 1 < λ0 < λ1 and let ‖ − ‖λ0, ‖ − ‖λ1 be thecorresponding seminorms on C∗(Γ) ⊗ Ad(Γ) introduced in (2.8) c). Thenthere exist constants C2, C3 such that the estimates

‖ ∇′(αn) ‖λ0 ≤ C2 ‖ αn ‖λ1

and

‖ (∇′ B′)k (αn) ‖λ0 ≤ (n + 1)(n + 3) · . . . · (n + 2k − 1) · Ck3· ‖ αn ‖λ1

hold for all αn ∈ Cn(Γ)⊗ Ad(Γ) and all n, k ∈ N.

Proof: The first assertion is an immediate consequence of (2.9) by noting

that for f0(t) := λt0, f1(t) := λt

1 the functionf ′0f1

is bounded on R+. Thesecond inequality of (2.9) yields for powers of the degree two operator∇′ B′ the estimate∥∥(∇′ B′)k(αn)

∥∥f≤ C0(n + 2k − 1)1−d

∥∥(∇′ B′)k−1(αn)∥∥

f ′

≤ Ck0(n + 2k − 1)1−d · (n + 2k − 3)1−d · . . . · (n + 1)1−d ‖ αn ‖ f (k)

so that ∥∥(∇′ B′)k(αn)∥∥

λ0≤

≤ Ck0((n + 2k − 1) · (n + 2k − 3) · . . .

·(n + 1))1−d supt≥0

(f (k)0

f1(t)

)‖ αn ‖λ1

166 M. Puschnigg

Here d, C0, C1 are the constants introduced in (2.9). Choose now a constantC such that exp

( dC0C

)< λ1

λ0and let C3 > exp

(dC0C1C

) · Cd · λC10 . Then

Ck0 · ((n + 2k − 1) · (n + 2k − 3) · . . . · (n + 1))−d ·

(f (k)0

f1(t)

)

≤ Ck0 ·(

1

k!)d

· (t + kC1)kd · λ(t+kC1)

0 · λ−t1

≤(

(C0 · C−1 · (t + kC1))k

k!)d

· Ckd · λ(t+kC1)0 · λ−t

1

≤(

exp

(C0

C· (t + kC1)

))d

· Ckd · λ(t+kC1)0 · λ−t

1

= exp

(d · C0

C· (t + kC1)

)· (Cd · λC1

0

)k ·(

λ0

λ1

)t

=(

exp

(dC0C1

C

)· Cd · λC1

0

)k

·(

exp

(dC0

C

)· λ0

λ1

)t

≤ Ck3

which establishes the second inequality.

3. Local cyclic homology of group Banach algebras

Hochschild and cyclic homology

To introduce notation a few well known facts about Hochschild and cyclichomology of algebras are collected in

Definition 3.1 [Co], [Co1]

a) For a complex algebra A define the A-bimodule of algebraic differentialforms by

Ωn A := A⊗ A⊗n ΩA :=⊕n

Ωn A

with A := A⊕C1 the algebra obtained from A by adjoining a unit. TheA-bimodule structure on ΩA is the obvious one.

b) The Hochschild complex of a complex algebra is given by

C∗(A) := (Ω∗A, b)

with Hochschild differential

b(a0 ⊗ . . .⊗ an) :=n−1∑i=0

(−1)ia0 ⊗ . . .⊗ aiai+1 ⊗ . . .⊗ an + (−1)nana0 ⊗ . . .⊗ an−1


Its homology HH∗(A, A) := H∗(C∗(A)) is called the Hochschild ho-mology of A. There is a canonical isomorphism

HH∗(A, A) Tor A⊗Aop

∗ (A, A)

c) The cyclic bicomplex of a complex algebra is the Z/2-graded chaincomplex

CC∗(A) :=(⊕

n

Ω∗+2n A, b+ B

)where the Connes differential B is given by

B(a0 ⊗ . . .⊗ an) :=n∑

j=0

(−1) j 1⊗ aj ⊗ . . .⊗ an ⊗ a0 ⊗ . . .⊗ aj−1

d) The Hodge-filtration of the cyclic bicomplex is the descending filtrationdefined by the subcomplexes

FilkHodge CC∗(A) :=

(bΩk A

⊕Ω≥k A, b + B

)generated by algebraic differential forms of degree at least k.

e) The periodic cyclic bicomplex CC∗(A) of a complex algebra is thecompletion of the cyclic bicomplex with respect to the Hodge filtration:

CC∗(A) := lim←n

CC∗/FilnHodgeCC∗(A)

Its homology HP∗(A) := H∗(CC∗(A)) is called the periodic cyclichomology of A.

f) The reduced Hochschild- and cyclic complexes of a unital algebra A aredefined similarly by using the A-bimodule ΩA of reduced algebraic dif-ferential forms. It is obtained from ΩA by adding the relation d(1A) = 0.The reduced Hochschild and cyclic complexes are naturally chain ho-motopic to the corresponding full complexes.

g) The bimodule of continuous differential forms over a complex Frechetalgebra A is the A-bimodule

Ωn Acont := A ⊗π A⊗nπ ΩAcont :=⊕

nΩn Acont

The continuous Hochschild, continuous cyclic and continuous periodiccyclic complexes of a Frechet algebra are defined similarly to the corres-ponding algebraic complexes by using continuous instead of algebraicdifferential forms.

Periodic cyclic (co)homology is a smooth homotopy functor and satisfiesexcision [Co], [CQ]. There exists a Chern character on the topologicalK-theory of Banach algebras with values in continuous periodic cyclichomology [Co].

168 M. Puschnigg

Cyclic homology of group rings

We collect some well known facts about the cyclic homology of group ringswhich can be found for example in [Bu] and [Ni].

For every group Γ there exists a commutative diagram of functors

Γ− BimodulesAd−−−→ Γ− Modules

−/[CΓ,−] (−)Γ

C− Vect C− Vectwhere the upper horizontal arrow associates to a Γ-bimodule M the samevector space M with the adjoint action and the vertical arrows are given bytaking the commutator quotient M → M/[CΓ, M] = M ⊗CΓ⊗CΓop CΓ re-spectively by taking coinvariants N → NΓ := N/〈n − gng−1|n ∈ N, g ∈Γ〉.The functor Ad is obviously exact and turns projective objects into projec-tive objects. From this one obtains a canonical isomorphism of functors

HH∗(−,CΓ) := TorCΓ⊗CΓop

∗ (−,CΓ) H∗(Γ, Ad(−))

which identifies the Hochschild-homology of CΓ with coefficients in a CΓ-bimodule M with the group homology with coefficients in Ad M. TakingM = CΓ one obtains a canonical isomorphism

HH∗(CΓ,CΓ) H∗(Γ, Ad(Γ))

This isomorphism can be described on the level of the projective standardresolutions, the bar resolutions. Moreover, Connes’ cyclic differential Bcorresponds to an explicit operator on the bar complex (C∗(Γ)⊗ Ad(Γ))Γ.This allows to describe the cyclic homology of group rings in terms of grouphomology. The formulas necessary for our purpose are summarized in

Lemma 3.2 [Ni] Let Γ be a group and let C∗(CΓ,CΓ) be the Hochschildcomplex of its complex group ring CΓ. Let C∗(Γ) ⊗ Ad(Γ) be the barresolution of Γ twisted by the adjoint representation and let (C∗(Γ) ⊗Ad(Γ))Γ be its complex of Γ-coinvariants.

a) There exist canonical isomorphisms

(C∗(Γ)⊗ Ad(Γ))Γµ

C∗(CΓ,CΓ) ∼= Ω∗(CΓ) .ν

inverse to each other. They are given on generators by the formulas

µ(g0, . . . , gn, v) =(g−1

n vg0)d(g−1

0 g1)d(g−1

1 g2) · · · d(g−1

n−1gn)

ν(g0dg1 . . . dgn) = (1, g1, g1g2, . . . , g1 . . . gn, g1 . . . gng0) .

These isomorphisms descend to isomorphisms

(C∗(Γ)⊗ Ad(Γ))Γµ

C∗(CΓ,CΓ) ∼= Ω∗(CΓ)

ν

of the corresponding reduced complexes.


b) The operator B′ : C∗(Γ) ⊗ Ad(Γ) → C∗+1(Γ) ⊗ Ad(Γ) of (2.6) isΓ-equivariant and descends therefore to an operator on the complex ofΓ-coinvariants. It corresponds under the isomorphisms in a) to Connes’differential B. A similar statement holds for the reduced complexes.

c) The operator ∇′ : C∗(Γ) ⊗ Ad(Γ) → C∗+1(Γ) ⊗ Ad(Γ) of (2.6) isΓ-equivariant and descends therefore to an operator on the complex ofΓ-coinvariants. It corresponds under the isomorphisms in a) to a con-nection ∇ on Ω

n(CΓ) for n ≥ d in the sense of Cuntz-Quillen [CQ1].

Proof: Statements a) and b) are straightforward and c) is a tautology becauseconnections in the sense of Cuntz and Quillen are by definition contractinghomotopies of the Hochschild complex.

A characteristic feature of the cyclic homology of group rings andcrossed products is described in

Lemma 3.3 [Ni] (Homogeneous decomposition)

a) The Γ-module Ad(Γ) is the direct sum of irreducible Γ-modules

Ad(Γ) =⊕〈γ 〉

Ad(Γ)〈γ 〉

labeled by the conjugacy classes 〈γ 〉 of Γ where Ad(Γ)〈γ 〉 is the linearspan of the elements of 〈γ 〉.

b) The decomposition of Ad(Γ) into irreducible submodules induces a cor-responding decomposition of (C∗(Γ) ⊗ Ad(Γ))Γ and therefore of theHochschild-complex C∗(CΓ,CΓ). It is compatible with Connes’ dif-ferential B and provides thus a decomposition of the cyclic bicomplexCC∗(CΓ). A similar statement holds for the reduced complexes.

c) The decomposition of b) is given explicitely by

CC∗(CΓ) =⊕〈γ 〉

CC∗(CΓ)〈γ 〉

where CC∗(CΓ)〈γ 〉 is the span of all algebraic differential formsg0dg1 . . . dgn with g0 · . . . · gn ∈ 〈γ 〉. The direct summand

CC∗(CΓ)hom = CC∗(CΓ)〈e〉

corresponding to the conjugacy class of the unit is called the homoge-neous part of CC∗(CΓ) whereas

CC∗(CΓ)inhom =⊕〈γ 〉=〈e〉

CC∗(CΓ)〈γ 〉

is called the inhomogeneous part.d) A similar decomposition holds for the Hochschild- and cyclic complexes

of crossed product algebras.

170 M. Puschnigg

e) The homogeneous decomposition of the Hochschild- and cyclic com-plexes induces a homogeneous decomposition of Hochschild- and cyclic(co)homology. In particular

HH∗(CΓ,CΓ)hom H∗(Γ,C).

Local cyclic homology

Local cyclic homology is a cyclic homology theory for topological alge-bras, for example Banach algebras. It is the target of a Chern character fromtopological K-theory and possesses properties similar to the K-functor, inparticular continuous homotopy invariance, excision property and stabil-ity [Pu1], [Pu2]. In addition it commutes with topological direct limits andis stable under passage to smooth dense subalgebras for Banach algebraswith metric approximation property [Pu1].

The local cyclic homology of a Banach algebra is given by an inductivelimit of cyclic type homology groups of dense subalgebras which makes itto some extent computable. We describe now its definition.

Let A be a Banach algebra and let U be its open unit ball. For eachcompact subset K ⊂ U denote by AK the completion of the subalgebraof A generated by K in the largest submultiplicative seminorm for which‖ K ‖≤ 1. In particular AK is a Banach algebra. For an auxiliary Banachalgebra A′ denote by ‖ − ‖N,m , N ≥ 1, m ∈ N the largest seminorm onΩA′ satisfying

‖ a0da1 . . . dan ‖N,m ≤ 1

c(n)!(2+ 2c(n))m N−c(n) ‖ a0 ‖A′ · . . . · ‖ an ‖A′

with c(2n) = c(2n + 1) = n. Let ΩA′(N) be the completion of ΩA′ withrespect to the seminorms ‖ − ‖N,m , m ∈ N. In fact Ωn A′(N) is then justa projective tensor power of A′ and ΩA′(N) becomes a weighted topologicaldirect sum of the subspaces Ωn A′(N). The cyclic differentials b and B extendto bounded operators on ΩA′(N). (This would not hold after completionwith respect to only one of the seminorms introduced above.) Denote byCC∗(A′)(N) the cyclic bicomplex CC∗(A′)(N) := (ΩA′(N), b+ B).

Definition 3.4 [Pu1] Let A be a Banach algebra. Then in the notationsintroduced above the local cyclic homology of A is defined as the directlimit

HCloc∗ (A) := lim−→

K⊂UN→∞

H∗(CC∗(AK )(N))

Recall that a Banach space E possesses the metric approximation prop-erty if the finite rank operators are dense in L(E) with respect to the


compact open topology and that a Banach algebra possesses this propertyif its underlying Banach space does.

For Banach algebras with metric approximation property the local cyclichomology can be calculated as a countable direct limit. Instead of runningover all compact subsets of the open unit ball it suffices to take the limitover the countable family of compact subsets K j := Vj ∩ B

( jj+1

)where

0 ⊂ V0 ⊂ . . . ⊂ Vj ⊂ . . . ⊂ A is an increasing sequence of finitedimensional subspaces such that

⋃j Vj is a dense subalgebra of A and

B( j

j+1

)is the closed j

j+1 -ball in A.

Proposition 3.5 [Pu1] Let A be a Banach algebra with metric approxi-mation property. Then in the notations above

HCloc∗ (A) := lim

j→∞N→∞

H∗(CC∗(AK j )(N))

Proof: This is [Pu1], (4.2).

Local cyclic homology of group Banach algebras

In the sequel the notion of a formal inductive limit or Ind-object will beused. Recall that for a category C the category Ind C of Ind-objects or formalinductive limits over C is defined as follows.

The objects of Ind C are small directed diagrams over C:

ob Ind C = “ lim−→I

”Ai

∣∣ I a partially ordered directed set

= Ai , fij : Ai → A j , i ≤ j ∈ I∣∣ f jk fij = fik

The morphisms between two Ind-objects are given by

morInd C(“ lim−→I

”Ai , “ lim−→J

”B j) := lim←−I

lim−→J

morC(Ai , B j)

where the limits on the right hand side are taken in the category of sets.Let Γ be a discrete group and let 1(Γ) be the convolution algebra of

summable functions on Γ. The Banach algebra 1(Γ) possesses the metricapproximation property. Suppose that Γ is finitely generated and let S bea finite symmetric set of generators. A natural choice for an increasing chain(Vj), j ∈ N of finite dimensional subspaces of 1(Γ) with dense union is totake Vj as the linear span of all elements g ∈ Γ with word length lS(g) ≤ j.Let K j be the corresponding family of compact subsets of 1(Γ) as definedabove. We describe the inductive family of Banach algebras 1 (Γ)K j , j ∈ N,explicitely. In order to do this we recall

172 M. Puschnigg

Definition 3.6 (Bost) [Bo] Let (Γ, S) be a finitely generated group anddenote for λ ≥ 1 by 1

λ(Γ) the completion of CΓ with respect to the largestseminorm satisfying

‖ g ‖≤ λlS(g)

The family 1λ(Γ), λ ≥ 1 is the inductive system of the Banach algebras

of summable functions on Γ of exponential decay in the word metric and1

1(Γ) = 1(Γ).

Lemma 3.7 For a finitely generated group (Γ, S) there is a natural iso-morphism

“ limj→∞

” 1(Γ)K j “limλ→1λ>1

” 1λ(Γ)

of Ind-Banach algebras. Moreover these are independent of the choice ofthe set of generators S.

Corollary 3.8 For a finitely generated group (Γ, S)

HCloc∗ (1(Γ)) lim

λ→1,λ>1N→∞

H∗(CC∗

(1

λ(Γ))(N)

)Proof: This follows from (3.5) and (3.7).

The norms on CC∗(1λ(Γ))(N) are given by weighted 1-norms on unions

of powers of Γ. An immediate consequence of this is

Lemma 3.9 Let 1(Γ) be the Banach algebra of a finitely generated group Γ.Then HCloc∗ (1(Γ)) possesses a homogeneous decomposition similar to theone described in (3.3).

It should be observed that there is no reason for the existence of a ho-mogeneous decomposition of HCloc∗ (CΓ) for completions CΓ of CΓ withrespect to other norms than weighted 1-norms.

The calculation of the local cyclic homology of 1(Γ) for a word hyper-bolic group will now be done in two steps.

In the first it is shown that the subcomplexes FilnHodgeCCloc∗ (1(Γ))

given by the Hodge filtration of the cyclic bicomplex do not contributeto HCloc∗ (1(Γ)) for large n. This is equivalent to the statement that the localcyclic homology of 1(Γ) is isomorphic to the direct limit of the continuousperiodic cyclic homology groups of the Banach algebras 1

λ(Γ).

Proposition 3.10 Let Γ be a word hyperbolic group. Let d be an integer asintroduced in (2.5). Then the following holds.

a) The natural map of Ind-complexes

“ limλ→1,λ>1

N→∞”CC∗

(1

λ(Γ))(N)−→ “lim

λ→1λ>1

”CC∗/FildHodgeCC∗(1

λ(Γ))


is an isomorphism in the homotopy category of Ind-complexes of com-plete, locally convex vector spaces. Here CC∗(1

λ(Γ))(N) is the comple-tion of the reduced cyclic bicomplex of the Banach algebra 1

λ(Γ) with re-spect to the seminorms‖ − ‖N,m introduced in (3.4) and CC ∗/FildHodgeCCdenotes the quotient of the reduced continuous cyclic bicomplex by thesubcomplex given by the closure of the d-th step of the Hodge filtration(3.1).

b) In particular

HCloc∗ (1(Γ)) lim

λ→1λ>1

H∗(CC∗/FildHodgeCC∗

(1

λ(Γ)))

c) The isomorphisms of a) and b) are compatible with the harmonic decom-position (3.3).

Proof: Let (Γ, S) be a word hyperbolic group. Let ∇′ be the contracting Γ-equivariant homotopy operator of (2.6) on the twisted reduced bar resolution

C∗(Γ)⊗ Ad(Γ) for ∗ ≥ d and let ∇ : Ω∗CΓ→ Ω

∗+1CΓ be the connection

on Ω∗(CΓ) corresponding to ∇′ under the isomorphism of (3.2) a).

Denote by ‖ − ‖λ,N,m the seminorms ‖ − ‖N,m (3.4) on CC∗(1λ(Γ))(N).

They satisfy ‖ − ‖λ,N′,m≤ C(m, N, N ′) ‖ − ‖λ,N,0 for every pair 1≤N < N ′so that one can restrict to the norms labeled by (λ, N, 0) in order to checkthe continuity of a morphism of the Ind-complexes under consideration.

The results of (2.10) provide, via the identifications (3.2), the followingestimates:

If λ > λ′ > 1 are real numbers and if C2, C3 are the constants introducedin (2.10), then for every pair of real numbers 1 ≤ N ≤ N ′ and for allω ∈ Ω(CΓ) the estimates

‖ ∇(ω) ‖λ′,N′ ,0≤ C2 ‖ ω ‖λ,N,0

and

‖ (∇ B)k(ω) ‖λ′,N′,0≤(

2C3

N ′

)k

‖ ω ‖λ,N,0

hold.The estimates above show that for fixed (λ, N) and any λ′ satisfying

1 < λ′ < λ there exists N ′ such that the linear maps ∇ and η :=∑∞k=0 (−∇ B)k ∇ extend to bounded operators from CC∗(1

λ(Γ))(N)

into CC∗(1λ′(Γ))(N′).

In particular the operators ∇ and η define bounded endomorphisms ofthe Ind-complex

“ limλ→1,λ>1

N→∞”FildHodgeCC∗

(1

λ(Γ))(N)

174 M. Puschnigg

and in fact the operator η defines a contracting homotopy of this Ind-complexbecause ∇ is a contracting homotopy of the Hochschild complex in degreeslarger or equal to d. Let

p : “ limλ→1,λ>1

N→∞”CC∗

(1

λ(Γ))(N)−→ “ lim

λ→1,λ>1N→∞


λ(Γ))(N)

be the natural quotient map. There exists a bounded linear map

s : “ limλ→1,λ>1

N→∞”CC∗/FildHodgeCC∗

(1

λ(Γ))(N)−→ “ lim

λ→1,λ>1N→∞

”CC∗(1

λ(Γ))(N)

of Ind-complexes which splits the projection p. It is defined by s p :=Id − b ∇ on differential forms of degree d − 1 and by zero respectivelythe identity in degrees above respectively below d − 1. The operator s′ :=s + η(s∂ − ∂s), with ∂ := b+ B the differential of the cyclic bicomplex, isthen a bounded chain map of Ind-complexes. In fact it is a homotopy inverseof p. To see this note that p s′ = Id and that Id − s′ p = ∂η′ + η′∂ withη′ := η(Id − s′ p). Finally

“ limλ→1,λ>1

N→∞”CC∗/FildHodgeCC∗

(1

λ(Γ))(N)

“limλ→1λ>1


λ(Γ))

as the weight factor N plays no role if only differential forms of uniformlybounded degree are considered. The proof of a) is thus complete and b) isan immediate consequence of (3.8) and a).

For c) note that by definition the contracting homotopy operator ∇′ ofthe reduced twisted bar resolution of (2.6) is natural with respect to thechosen coefficient module and therefore compatible with the decompos-ition of Ad(Γ) into irreducible subspaces. By (3.2) this is equivalent to thecompatibility of the connection ∇ with the homogeneous decomposition ofCC∗(CΓ) from which c) follows along the lines of the proof of a).

The homology of the inductive system of truncated cyclic complexesof the Ind-Banach algebra “lim

λ→1λ>1

”1λ(Γ) is studied further by comparing it

with the twisted bar complexes and Rips chain complexes of the previoussection.

We calculate only the homogeneous part of HCloc∗ (1(Γ)) as this is whatwe need for the purpose of this paper. The inhomogeneous part will beconsidered elsewhere.

Theorem 3.11 Let Γ be a word-hyperbolic group. Then there exists a nat-ural isomorphism

HCloc∗ (1(Γ))hom H∗

(Γ, HCloc

∗ (C)) :=⊕

n

H∗+2n(Γ,C)

between the homogeneous part of the local cyclic homology of 1(Γ) andthe homology of Γ with two-periodic complex coefficients.


Proof: The homogeneous decomposition takes the form

HCloc∗ (1(Γ)) = HCloc

∗ (1(Γ))〈e〉⊕

HCloc∗ (1(Γ))inhom

so that by the results of (3.10)

HCloc∗ (1(Γ))hom lim

λ→1λ>1

H∗(CC∗/Fild

HodgeCC∗(1

λ(Γ))〈e〉)

Define a bicomplex C∗∗(Γ)(d)Γ by

C pq(Γ)(d)Γ :=

Cq−p(Γ)Γ 0 ≤ q − p < d − 1Cd−1/∂barCd(Γ)Γ q − p = d − 10 otherwise

with differentials d0 := ∂bar : C pq(Γ)(d)Γ → C p(q−1)(Γ)

(d)Γ the differential

of the bar complex and d1 := B′ : C pq(Γ)(d)Γ → C(p−1)q(Γ)

(d)Γ (2.6).

Denote by CR∗∗(Γ)

(d)Γ the corresponding bicomplex associated to the Rips

chain complex CR∗ (Γ)Γ ⊂ C∗(Γ)Γ (2.2). It is a subbicomplex of C∗∗(Γ)

(d)Γ .

Note that CR∗ (Γ)Γ and therefore also C

R∗∗(Γ)

(d)Γ are finite dimensional in

each degree.The natural morphisms of (3.2) induce an isomorphism of complexes

ν : CC∗/FildHodgeCC∗(CΓ)〈e〉

−→ Tot(C∗∗(Γ)

(d)Γ

)between the quotient of the homogeneous part of the cyclic bicomplex ofCΓ by the d-th step of the Hodge filtration and the total complex of thebicomplex C∗∗(Γ)

(d)Γ . After completion of C∗∗(Γ)

(d)Γ with respect to the

seminorms ‖ − ‖λ of (2.8) one obtains a similar isomorphism

ν : “limλ→1λ>1

”CC∗/FildHodgeCC∗

(1

λ(Γ))〈e〉

−→ “limλ→1λ>1

”Tot((

C∗∗(Γ)(d)Γ

)λ

)of Ind-complexes.

We claim that the inclusion

ι : Tot(C

R∗∗(Γ)

(d)Γ

) −→ “limλ→1λ>1

”Tot((

C∗∗(Γ)(d)Γ

)λ

)is a chain homotopy equivalence of Ind-complexes. Note that it suffices toprove that the inclusion is a chain homotopy equivalence on the columnsof the bicomplexes because the bicomplexes are concentrated in a strip of

176 M. Puschnigg

finite width. In fact the chain map Φ : C∗(Γ)Γ → C∗(Γ)Γ of (2.5) factorsthrough C

R∗ (Γ)Γ and by (2.9) gives rise to a morphism

Φ : “limλ→1λ>1

”((

C∗(Γ)(d)Γ

)λ

)→ “limλ→1λ>1

”((

CR∗ (Γ)

(d)Γ

)λ

) = CR∗ (Γ)

(d)Γ

because “limλ→1λ>1

”((CR∗ (Γ)

(d)Γ )λ) is a constant, finite dimensional Ind-complex.

The endomorphism ι Φ of “limλ→1λ>1

”((C∗(Γ)(d)Γ )λ) is chain homotopic to the

identity because Id− ιΦ = ∇′∂+∂∇′ and∇′ is a bounded operator on theconsidered Ind-complex by (2.10). On the other hand the chain endomor-

phism Φ ι of the finite dimensional complex CR∗ (Γ)

(d)Γ is chain homotopic

to the identity because Id − Φ ι is a chain map of the Rips resolutionwhich vanishes in degree zero and any such chain map is nullhomotopic.Altogether this implies that

HCloc∗ (1(Γ))hom H∗

(Tot(C

R∗∗(Γ)

(d)Γ

))Let π be the antisymmetrization operator on C

R∗ (Γ) (2.1) and denote

by CR∗∗(Γ)′Γ the bicomplex with C

Rpq(Γ)′Γ := π(C

Rq−p(Γ)Γ) and differentials

d0 = ∂bar and d1 = 0. Note that CRpq(Γ)′Γ = 0 for q− p ≥ d as π annihilates

CR∗ (Γ) for ∗ ≥ d. The antisymmetrization map defines then a map

π ′ : CR∗∗(Γ)

(d)Γ → C

R∗∗(Γ)′Γ

of bicomplexes because π B′ = 0 on C∗(Γ) ⊂ C∗(Γ)⊗ Ad(Γ).The map π ′ is a chain homotopy equivalence on columns as π is a map of

Rips-resolutions equal to the identity in degree zero. Therefore π ′ is a chainhomotopy equivalence of total complexes, too, and

HCloc∗ (1(Γ))hom

⊕n

H∗+2n(π(C

R∗ (Γ)

)Γ

)The Rips resolution C

R∗ (Γ) is a free resolution of the constant Γ-module C.

Its complex of Γ-coinvariants calculates therefore the homology of Γ withcomplex coefficients and the same holds for the image of the complex underantisymmetrization. This finally yields the isomorphism

HCloc∗ (1(Γ))hom

⊕n

H∗+2n(Γ,C)

claimed by the theorem. The notation H∗(Γ, HCloc∗ (C)) has been introducedto underline the analogy of the map above with the assembly map in K-theory.


Remark 3.12 In [Pu3] it was shown that Theorem (3.11) holds also forfundamental groups of compact nonpositively curved manifolds. In fact thehomogeneous part of HCloc∗ (1(Γ)) can be identified with certain homologygroups of Γ for a quite large class of groups. This together with an analysisof the inhomogeneous part will be presented elsewhere.

4. Auxiliary results about crossed products and their local cyclichomology

The rapid decay property of discrete groups

Let Γ be a finitely generated discrete group with finite symmetric set ofgenerators S. Denote the associated word length function by lS.

Let 2(Γ) be the Hilbert space of square intergrable functions on Γ.The group algebra CΓ acts as ∗-algebra of operators on 2(Γ) and both CΓand its closure C∗r (Γ) under the operator norm can be identified with linearsubspaces of 2(Γ) by associating to an operator the image of the cyclic andseparating vector e1 ∈ 2(Γ).

Definition 4.1 [Jol] Denote by A(Γ) the completion of CΓ with respect tothe family of seminorms

‖∑

aγ uγ‖2k :=

∑γ

(1+ ls(γ))2k|aγ |2, k ≥ 0

It is a linear subspace of 2(Γ) containing CΓ and is independent of thechoice of the finite generating set S. A finitely generated group Γ is said topossess the property of rapid decay (RD) if

A(Γ) ⊂ C∗r (Γ)

as subspaces of 2(Γ).

Proposition 4.2 [Jol] Let Γ be a finitely generated group which possessesthe property (RD) of rapid decay. Then A(Γ) is an admissible Frechet al-gebra which is closed under holomorphic functional calculus in the reducedgroup C∗-algebra C∗r Γ. It is called the Jolissaint algebra of Γ.

Proof: Let 2(Γ) be the Hilbert space with standard orthonormal basisξg, g ∈ Γ. Let F be the unbounded selfadjoint operator on 2(Γ) givenby F(ξg) := lS(g)ξg and let D be the unbounded derivation D := [F,−]on B(2(Γ)). The group ring CΓ acts by convolution as involutive algebraof bounded operators on 2(Γ). It satisfies CΓ ⊂ ⋂

ndom(Dn). Moreover

‖ Dn(a)(e1) ‖2H =

∑g∈Γ

lS(g)2n|ag|2 for a ∈ CΓ.

If Γ satisfies the property of rapid decay then by the closed graph theo-rem ‖ a ‖B(H )≤ C ‖ a ‖k for a ∈ A(Γ) and some constants C and k.

178 M. Puschnigg

A straightforward calculation shows then ‖ Dn(a) ‖B(H )≤ C ‖ a ‖n+kfor a ∈ A(Γ) and n ∈ N. Therefore A(Γ) coincides with the closure ofCΓ in

⋂n

dom(Dn) ⊂ B(2(Γ)), i.e. with the completion of CΓ with re-

spect to the seminorms ‖ a ‖′k:=‖ Dk(a) ‖B(2(Γ)). This implies that A(Γ)is a Frechet algebra and shows also that it is closed under holomorphicfunctional calculus in C∗r (Γ).

We claim that A(Γ) is admissible. This means that it possesses an “openunit ball” U such that the multiplicative closure of any compact subset ofU is relatively compact in the ambient algebra [Pu]. Our choice for an openunit ball is U := a ∈ A(Γ), ‖∑

γ

|aγ |uγ ‖C∗r (Γ) < 1. In fact let K ⊂ U be

compact, ‖ K ‖C∗r (Γ) ≤ ε < 1. Then

‖ Dk(Kn) ‖B(H )

≤∑

j1+...+ jm=ki1+...+im+1=n−k

‖ Ki1 ‖‖ D j1(K ) ‖‖ Ki2 ‖ · . . . · ‖ D jm(K ) ‖‖ Kim+1 ‖

≤ nk εn−k max0≤ j≤k

‖ D j(K ) ‖B(H )≤ C ′(k, K )

which shows that the multiplicative closure of K is bounded with respectto all the defining seminorms of the Jolissaint algebra and thus a relativelycompact subset of A(Γ). This proves our claim.

The homogeneous decomposition for good completions of crossed products

Following Lafforgue [La] we introduce the notion of a good completion ofa group algebra inside its reduced group C∗-algebra. We use an argumentdue to Connes and Moscovici [CM] to show that the local cyclic homologygroups of any good completion of CΓ possess a homogeneous decompos-ition and that the homogeneous parts coincide for any two sufficiently largegood completions.

Definition 4.3 [La] Let Γ be a discrete group. A good completion of thegroup ring CΓ is an admissible Frechet subalgebra a(Γ) of C∗r (Γ) whichcan be defined by seminorms satisfying the conditions

‖∑

γ

aγ uγ ‖=‖∑

γ

|aγ |uγ ‖

and

|aγ | ≤ |bγ | , ∀γ ∈ Γ ⇒‖∑

γ

aγ uγ ‖≤‖∑

γ

bγ uγ ‖

A good completion of CΓ is called sufficiently large if it contains all theBanach algebras 1

λ(Γ), λ > 1, (3.6) of summable functions of exponentialdecay.


Examples of good completions are the group Banach algebra 1(Γ) andthe Jolissaint algebra of a group with the property of rapid decay. Stillfollowing Lafforgue one has

Lemma 4.4 [La] Let Γ be a discrete group and let a(Γ) be a good com-pletion of CΓ with defining seminorms ‖ − ‖k, k ∈ N. Let A be a Banach-algebra with isometric Γ-action.

a) The completion a(Γ, A) of the algebraic crossed product A Γ withrespect to the seminorms

‖∑

γ

aγ uγ‖′ k := ‖∑

γ

‖aγ‖A uγ ‖k

is an admissible Frechet algebra.b) If A is a Γ-C∗-algebra then a(Γ, A) is a dense Frechet subalgebra of

the reduced crossed product C∗-algebra A r Γ.

Proof: Let A be a Banach algebra with isometric Γ-action.For a = ∑

aγ uγ ∈ a(Γ, A) put |a| := ∑ ‖aγ‖A uγ ∈ a(Γ). It followseasily from the definitions and the identity ‖a‖a(Γ,A)

k = ‖ |a| ‖a(Γ)k that

a(Γ, A) is a Frechet algebra and is admissible [Pu], an open unit ball beinggiven by U := a ∈ a(Γ, A), |a| ∈ U ′ where U ′ is an open unit ball ofa(Γ).

Suppose now that A is a Γ-C∗-algebra which is represented faithfullyon the Hilbert space H A and consider the associated representation ofAred Γ on H A⊗2(Γ). A simple calculation shows that ‖ a ‖B(H A⊗2(Γ))≤‖ |a| ‖C∗r (Γ) which implies a(Γ, A) ⊂ Ared Γ.

In particular the local cyclic homology groups of good completions ofcrossed products are well defined as these are admissible Frechet algebras.

A crucial step in the proof of the Novikov conjecture for word-hyperbolicgroups by Connes and Moscovici [CM] is the observation that every con-tinuous homogeneous cyclic cocycle on the group Banach algebra 1(Γ) ofa finitely generated discrete group extends to a continuous cyclic cocycle onany good completion of CΓ. This is somewhat surprising because there isa priori no relation between the cyclic groups of different good completions.Repeating the argument of Connes and Moscovici in our context we find

Proposition 4.5 (after Connes and Moscovici) Let Γ be a finitely generateddiscrete group and let a(Γ) be a sufficiently large good completion of CΓ(4.3). Let A be a Banach algebra with isometric Γ-action.

a) The local cyclic homology groups of a(Γ, A) possess a homogeneousdecomposition

HCloc∗(a(Γ, A)

) HCloc∗(a(Γ, A)

)hom⊕ HCloc

∗(a(Γ, A)

)inhom

similar to the one in (3.2).

180 M. Puschnigg

b) There exists a natural isomorphism

HCloc∗(1(Γ, A)

)hom

−→ HCloc∗ (a(Γ, A)

)hom

of the homogeneous parts of local cyclic homology, i.e. the homogeneouspart is independent of the choice of a sufficiently large good completion.

Proof: The proof is that of [CM], (6.5). We begin with some preliminar-ies. The notations of (3.4) and (3.5) are permanently used. The inclusions1

λ(Γ) → 1λ′(Γ), λ > λ′ > 1 are compact so that by assumption the same

holds for the inclusions 1λ(Γ) → a(Γ). In particular one obtains a natural

bounded morphism η : “limλ→1λ>1

” 1λ(Γ) −→ “ lim

K⊂U” a(Γ)K of Ind-Banach-

algebras. This induces a morphism of cyclic Ind-complexes which yieldsthe canonical homomorphism η∗ : HCloc∗ (1(Γ)) −→ HCloc∗ (a(Γ)) of localcyclic homology groups.

Let U be an open unit ball for a(Γ) given by an open ball with respectto some seminorm satisfying the conditions of (4.3) and let K j ⊂ a(Γ)

be the compact set K j := Vj ∩ jj+1U where Vj is the linear span of the

set of elements of Γ of word length at most j. We claim that there existsλ > 1 (depending on j) such that the linear endomorphism of CΓ given byug → λlS(g) ug extends to a bounded linear map ιλ : a(Γ)K j → a(Γ). Infact this is true provided that λ > 1 is such that λ j · K j ⊂ U as a simplecalculation shows.

Let πhom be the canonical projection onto the homogeneous part ofCC∗(CΓ). Its restriction to (CΓ)⊗n+1

is given by the formula

πhom : (CΓ)⊗n+1 −−−→ (CΓ)⊗n+1

a0 ⊗ . . .⊗ an −−−→ ∑γ0...γn=e

(aγ0uγ0)0 ⊗ . . .⊗ (aγn uγn)

n

Let j ∈ N and suppose that λ > 1 is chosen so that ι = ιλ extends toa bounded operator as described above. Then

‖ πhom(a0 ⊗ . . .⊗ an) ‖1λ(Γ)⊗n+1

π≤

∑γ0...γn=e

(λlS(g0)|aγ0 |) . . . (λlS(gn)|aγn |)

= | 〈ξe, |ι(a0)| · . . . · |ι(an)|ξe〉 | ≤ ‖ |ι(a0)| ‖C∗r (Γ) . . . ‖ |ι(an)| ‖C∗r (Γ)

≤ Cn+1 ‖ ι(a0) ‖k . . . ‖ ι(an) ‖k

for some constant C and some seminorm ‖ − ‖k by the closed graph theorem

≤ C( j, k)n+1 ‖ a0 ‖a(Γ)K j. . . ‖ an ‖a(Γ)K j

This suffices to verify that πhom defines a bounded chain map of Ind-complexes

π ′hom : “ limj→∞N→∞

” CC∗(a(Γ)K j )(N) −→ “ limλ→1,λ>1

N→∞” CC∗(1

λ(Γ))(N)


Good completions of CΓ possess the metric approximation property. Soby the approximation theorem (3.5) their local cyclic homology can becalculated by the Ind-complex considered above. Together with (3.8) itfollows that this chain map considered above induces a natural homo-morphism

π ′hom∗ : HCloc∗ (a(Γ)) −→ HCloc

∗ (1(Γ))hom

of local cyclic homology groups. It is clear that the endomorphism

πhom∗ := η∗ π ′hom∗ : HCloc∗ (a(Γ)) −→ HCloc

∗ (a(Γ))

is idempotent and defines a homogeneous decomposition of HCloc∗ (a(Γ)).Moreover it follows from the continuity of π ′hom that the restriction of η∗ tothe homogeneous part

η∗ : HCloc∗ (1(Γ))hom −→ HCloc

∗ (a(Γ))hom

is an isomorphism. The extension of this to the case of crossed products isstraightforward.

Before we proceed a lemma about the compatibility of homogeneousdecompositions with boundary maps in long exact homology sequences isneeded.

Lemma 4.6 Let

0 → I → Af→ B → 0

be a Γ-equivariant extension of Γ-algebras with (not necessarily equivari-ant) linear section. Then the long exact periodic cyclic homology sequenceassociated to the extension

0 → I Γ→ A ΓfΓ−→ B Γ→ 0

of crossed products decomposes into the direct sum of long exact sequencesof the homogeneous parts

. . .→ HP∗(A Γ)homfΓ−→ HP∗(B Γ)hom

δ−→ HP∗−1(I Γ)hom → . . .

and the inhomogeneous parts

. . .→ HP∗(A Γ)inhomfΓ−→ HP∗(B Γ)inhom

δ−→ HP∗−1(I Γ)inhom → . . .

182 M. Puschnigg

Proof: The inclusions

CC∗(I Γ) ⊂ CC∗(A Γ, B Γ) ⊂ Cone( f Γ)∗

of bicomplexes are compatible with the homogeneous decomposition. Theexcision theorem in periodic cyclic homology [CQ] states that the inclusionsabove induce isomorphisms on homology. The same has to be true then forthe inclusions of the homogeneous respectively inhomogeneous parts of thebicomplexes which is equivalent to the assertion of the lemma. Lemma 4.7 Let Γ be a finitely generated group and let a(Γ) be a sufficientlylarge good completion of CΓ. Let

0 → I → A → B → 0

be an equivariant extension of Banach-algebras with isometric Γ-actionwhich possesses a (not necessarily Γ-equivariant) bounded linear section.

Then the long exact sequence of local cyclic homology groups of theextension

0 → a(Γ, I ) → a(Γ, A) → a(Γ, B)→ 0

decomposes into the direct sum of long exact sequences of the homogeneousrespectively inhomogeneous parts.

Proof: The Ind-complexes which define the local cyclic homology (3.4) ofa(Γ,−) possess a homogeneous decomposition by (4.5). The extensionsof crossed products above have an obvious bounded linear section derivedfrom the section of the original extension. The assertion follows then bythe same arguments as in the proof of (4.6) from the excision theorem inlocal cyclic homology for extensions of admissible Frechet algebras [Pu2].

The homogeneous decomposition for the local cyclic homology of some

crossed products is determined now. We are interested in crossed productsof Γ-C∗-algebras over proper and free Γ-spaces.

Proposition 4.8 Let Γ be a torsion-free finitely generated group and leta(Γ) be a sufficiently large good completion of CΓ. Let X be a simplicialcomplex on which Γ acts freely and simplicially and suppose that X := X/Γis a finite complex. Let A be a Γ-C0(X)-C∗-algebra [Ka2], (1.5).

Then the canonical homomorphisms

HCloc∗ (a(Γ, A))

−→ HCloc∗ (C∗r (Γ, A))

and

HCloc∗ (a(Γ, A))hom

−→ HCloc∗ (a(Γ, A))

are isomorphisms of local cyclic homology groups.


Proof: Recall that a Γ-C0(X)-C∗-algebra is a Γ-C∗-algebra with a Γ-equivariant structure homomorphism C0(X) → Z(M(A)) of C0(X) to thecenter of the multiplier algebra of A which satisfies lim−→ una = a, ∀a ∈ A,

for some bounded approximate unit (un) of C0(X).

Let ∆ be a topdimensional simplex of X and put Y := X − ∆. Y is

a simplicial complex with one simplex less than X. Denote by π : X → X

the canonical projection and put Y := π−1Y, ∆ := π−1(∆). Then

0 → C0(∆)→ C0(X) → C0(Y )→ 0

is a Γ-equivariant extension of Γ-C0(X)-C∗-algebras with equivariantbounded linear section. The sequence

0 → C0(∆)⊗C0(X) A → C0(X)⊗C0(X) A → C0(Y)⊗C0(X) A → 0

(see [Ka2] (1.6)) is again an extension of Γ-C∗-algebras with boundedlinear section. Note that C0(X)⊗C0(X) A = A and that I := C0(∆)⊗C0(X) A

respectively B := C0(Y) ⊗C0(X) A is a Γ-C0(∆)- respectively a Γ-C0(Y )-algebra.

The action of Γ on X being free the maximal and reduced crossed prod-ucts of any Γ-C0(X)-C∗-algebra with Γ agree. It follows that the sequenceof reduced crossed products associated to the extension 0 → I → A →B → 0 is exact because the corresponding sequence of maximal crossedproducts is always exact. In particular there exists a natural map of longexact sequences

δ→ HCloc∗ (C∗r (Γ, I ))→ HCloc∗ (C∗r (Γ, A)) → HCloc∗ (C∗r (Γ, B))δ→

↑ ↑ ↑δ→ HCloc∗ (a(Γ, I )) → HCloc∗ (a(Γ, A)) → HCloc∗ (a(Γ, B))

δ→By the results of (4.7) there exists also a natural map of long exact sequences

δ→ HCloc∗ (a(Γ, I )) → HCloc∗ (a(Γ, A)) → HCloc∗ (a(Γ, B))δ→

↑ ↑ ↑δ→ HCloc∗ (a(Γ, I ))hom → HCloc∗ (a(Γ, A))hom → HCloc∗ (a(Γ, B))hom

δ→We argue now by induction over the number of simplices of X. The twodiagrams above allow with the help of the five lemma to reduce to the casethat X is a single (open or closed) simplex.

Under this condition a Γ-C0(X)-algebra is necessarily of the formA C0(Γ)⊗ A′ for a C∗-algebra A′ with trivial Γ-action.

Let S ⊂ Γ be a finite subset, let χS ∈ Cc(Γ) be its characteristic functionand denote by PS ∈ Cc(Γ)Γ the image of χS under the canonical inclusion.One verifies easily that PS is a multiplier of (Cc(Γ)⊗ A′) Γ and that the

184 M. Puschnigg

maps πS : (Cc(Γ)⊗ A′)Γ→ (Cc(Γ)⊗ A′)Γ, πS(α) := PS ·α · PS ex-tend to idempotent, norm decreasing linear endomorphisms of a(Γ, A) andC∗r (Γ, A) satisfying lim−→

S⊂Γ

πS = Id pointwise. Moreover the images of the

maps πS are closed subalgebras. It is well known that πS((Cc(Γ)⊗ A′)Γ) Mn(A′) for n = #S. In particular πe((Cc(Γ) ⊗ A′) Γ) A′. Thestability under passage to matrix algebras and the approximation theoremfor local cyclic homology [Pu1], (4.2) imply then that the maps

HCloc∗ (A′)→ lim−→

S⊂Γ

HCloc∗ (πS(a(Γ, A))) → HCloc

∗ (a(Γ, A))

and

HCloc∗ (A′)→ lim−→

S⊂Γ

HCloc∗ (πS(C

∗r (Γ, A))) → HCloc

∗ (C∗r (Γ, A))

are isomorphisms from which the first assertion follows. It is also clear byconstruction that the first isomorphism above factors as

HCloc∗ (A′)→ HCloc

∗ (a(Γ, A))hom → HCloc∗ (a(Γ, A))

which shows the second assertion.

5. Equivariant Chern-Connes characters and the Kadison-Kaplanskyconjecture

Equivariant Chern-Connes characters

The description of Kasparov’s bivariant K-theory [Ka] in terms of universalalgebras by Cuntz [Cu] allows to characterize KK-theory as the universalstable and split exact homotopy functor on the category of (separable)C∗-algebras [Hi]. Recently Thomsen gave a similar characterization ofequivariant KK-theory.

Theorem 5.1 [Th] Let Γ be a locally compact second countable group. LetKKΓ be the additive category with separable Γ-C∗-algebras as objects andmorphisms defined by morKKΓ(A, B) := KKΓ(A, B). The composition ofmorphisms is given by the Kasparov product. Let ι be the tautological functorfrom the category of separable Γ-C∗-algebras to KKΓ which assigns to aΓ-homomorphism f : A → B the element f∗ ∈ KKΓ(A, B). Then everystable and split exact homotopy functor F from the category of separableΓ-C∗-algebras to an additive category C factors uniquely through KKΓ,i.e. there exists a unique functor F :KKΓ −→ C such that F = F ι.

We derive, similar to [Cu1], a number of consequences from this result.

Theorem 5.2 Let Γ be a finitely generated discrete group and let a(Γ) bea sufficiently large good completion (4.3) of the group ring CΓ.


There exist natural transformations of bifunctors,

chΓbiv : KKΓ(−,−) −→ HCloc

0 (−r Γ,−r Γ)

and

cha(Γ)biv : KKΓ(−,−) −→ HCloc

0 (a(Γ,−), a(Γ,−))

called the equivariant Chern-Connes characters, on the category of sep-arable Γ-C∗-algebras. They satisfy the following conditions:

• If α ∈ KKΓ(A,B) is given by a Γ-equivariant homomorphism f : A→Bof C∗-algebras then

chΓbiv (α) = ( f Γ)∗ ∈ HCloc

0 (A r Γ, B r Γ)

and

cha(Γ)biv (α) = ( f Γ)∗ ∈ HCloc

0 (a(Γ, A), a(Γ, B))

• The equivariant Chern-Connes characters are multiplicative, i.e. forany separable Γ-C∗-algebras A, B, C and for any α ∈ KKΓ(A, B) andβ ∈ KKΓ(B, C)

chΓbiv(α β) = chΓ

biv(α) chΓbiv(β)

and

cha(Γ)biv (α β) = cha(Γ)

biv (α) cha(Γ)biv (β)

In fact these properties characterize the equivariant Chern-Connes char-acters uniquely.

• There exist canonical natural transformations of bifunctors

jr : KKΓ(−,−) −→ KK(−r Γ,−r Γ)

and

ja : KKΓ(−,−) −→ Hom(K∗(a(Γ,−)), K∗(a(Γ,−)))

which are compatible with products. For any α ∈ KKΓ(A, B) the dia-gram

K∗(a(Γ, A))ja(α)−−−→ K∗(a(Γ, B))

K∗(Ar Γ)jr(α)−−−→ K∗(B r Γ)

commutes.

186 M. Puschnigg

• The equivariant Chern-Connes character is compatible with the Cherncharacter: for any α ∈ KKΓ(A, B) the diagrams

K∗(Ar Γ)jr(α)−−−→ K∗(B r Γ)

ch

ch

HCloc∗ (A r Γ)chΓ

biv(α)−−−−→ HCloc∗ (B r Γ)

and

K∗(a(Γ, A))ja(α)−−−→ K∗(a(Γ, B))

ch

ch

HCloc∗ (a(Γ, A))cha(Γ)

biv (α)−−−−→ HCloc∗ (a(Γ, B))

commute.• The equivariant Chern-Connes character is compatible with the homo-

geneous decomposition of local cyclic homology: for any α ∈ KKΓ(A,B)the morphism

HCloc∗ (a(Γ, A))cha(Γ)

biv (α)−−−−→ HCloc∗ (a(Γ, B))

preserves the homogeneous decomposition, i.e. it decomposes into thedirect sum of the morphisms

HCloc∗ (a(Γ, A))homcha(Γ)

biv (α)−−−−→ HCloc∗ (a(Γ, B))hom

of the homogeneous and

HCloc∗ (a(Γ, A))inhomcha(Γ)

biv (α)−−−−→ HCloc∗ (a(Γ, B))inhom

of the inhomogeneous parts.

Proof: The assertions of the theorem are immediate consequences of thecharacterization of equivariant KK -theory in (5.1).

Introduce a functor F on the category of separable Γ-C∗-algebras byF(A) := HCloc∗ (a(Γ, A)) and F( f ) := ( f Γ)∗ ∈ HCloc

0 (a(Γ, A), a(Γ, B))for f : A → B a Γ-homomorphism. F is viewed as functor with values inthe category C with objects given by the image of F and with morphismsgiven by

morC(HCloc∗ (a(Γ, A)), HCloc

∗ (a(Γ, B))) := HCloc0 (a(Γ, A), a(Γ, B))

The composition of morphism is given by the composition product. Localcyclic homology is a stable and split exact homotopy functor [Pu1], [Pu2].It follows therefore from Thomsen’s theorem (5.1) that F factors through


KKΓ. The corresponding functor F : KKΓ → C induces then theequivariant Chern-Connes character cha(Γ)

biv on morphisms. By construc-tion cha(Γ)

biv ( f∗) = F( f∗) = F( f ) = ( f Γ)∗ and the multiplicativity ofcha(Γ)

biv is equivalent to the statement that F is a functor. The construction ofchΓ

biv is similar. The uniqueness is obvious.Introduce a functor F ′ on the category of separable Γ-C∗-algebras with

values in the category KK (defined similarly to the equivariant categories in(5.1)) by F ′(A) := K∗(ArΓ) and F ′( f ) := ( f Γ)∗ ∈ KK(ArΓ, BrΓ)for f : A → B a Γ-homomorphism. Introduce still another functor F ′′ onthe category of separable Γ-C∗-algebras with values in a full subcategoryof the category of Z/2Z-graded abelian groups by F ′′(A) := K∗(a(Γ, A))and F( f ) := ( f Γ)∗. As K -theory is a stable and split exact homotopyfunctor the functors F ′ and F ′′ factor again by Thomsens theorem throughKKΓ giving rise to the transformations jr and ja. Their compatibilitywith products is again equivalent to the fact that F ′ and F ′′ are functors.The natural homomorphism K∗(a(Γ, A)) → K∗(A r Γ) induced by theinclusion of algebras a(Γ, A) → A r Γ defines a natural transformationF ′ → F ′′ of functors. It gives rise to a natural transformation F ′ → F ′′which proves the desired commutativity of the diagram relating jr and ja.

The Chern character ch : K∗ → HCloc∗ in local cyclic homology de-fines natural transformations between the stable and split exact homotopyfunctors F ′ and F respectively F ′′ and F introduced above. The associatednatural transformations ch : F ′ → F respectively ch : F ′′ → F of functorsfrom KKΓ to Ab yield the compatibility of the ordinary Chern characterwith the equivariant Chern-Connes character claimed in the theorem.

Consider the functors Fhom and Finhom on the category of separable Γ-C∗ -algebras given by Fhom (A) := HCloc

0 (a(Γ, A))hom respectively Finhom (A) :=HCloc

0 (a(Γ, A))inhom and Fhom( f ) = Finhom ( f ) = ( f Γ)∗. They are welldefined because the homogeneous decomposition of HCloc(a(Γ, A)) existsby (4.5) and is preserved by the elements of HCloc

0 (a(Γ, A), a(Γ, B)) of theform ( f Γ)∗. As both functors are stable, split exact and homotopy invari-ant by [Pu1] and (4.7), the direct sum decomposition F Fhom ⊕ Finhomgives rise to a direct sum decomposition F Fhom ⊕ Finhom of functorsfrom KKΓ to Ab which proves the last assertion of the theorem.

Assembly maps and the γ -element

Let Γ be a torsion-free discrete group. Let M be a smooth compact mani-fold without boundary and let D be an elliptic differential operator actingon the sections of a vector bundle over M. Let M be a Γ-covering of M andlet D be the lift of D to a Γ-invariant elliptic differential operator actingon the sections of the appropriate Γ-equivariant vector bundle on M. Kas-parov [Ka1] associates to the data (Γ, M, D) on the one hand a topologicalindex Indt(D) ∈ Ktop

0 (BΓ) in the topological K-homology group of the clas-sifying space of Γ. In fact the topological indices exhaust the group K0(BΓ).

188 M. Puschnigg

On the other hand he defines an analytic index Inda(D) ∈ K0(C∗r (Γ)) in theK-group of the reduced group C∗-algebra of Γ. For the trivial group thesequantities coincide with the topological and analytic indices of an ellipticoperator on a compact manifold introduced by Atiyah and Singer [AS].There exists a natural assembly homomorphism

µ : Ktop∗ (BΓ) −→ K∗(C∗r (Γ))

∈ ∈

Indt(D) −→ Inda(D)

[BC] which maps the topological index of an elliptic operator D to itsanalytic index.

The index theorem of Atiyah-Singer states that the topological and an-alytic index of an elliptic operator are identical. In particular the groups inwhich topological and analytic indices live are canonically isomorphic. Themost optimistic hope for a generalization in the context of Γ-index theorywould be that the assembly map µ is an isomorphism. This is the contentof the famous Baum-Connes Conjecture [BC], [BCH].

All the approaches to this conjecture known up to now use in one oranother form a strategy formulated by Kasparov [Ka1], [Ka2]. It is basedon the construction of so called γ -elements and important for us becausethese elements provide a canonical decomposition of K∗(C∗r (Γ)) into thedirect sum of the image and the cokernel of the assembly map.

Definition 5.3 [Ka1], [Ka2] Let Γ be a torsion-free discrete group andsuppose that the classifying space BΓ has the homotopy type of a finitedimensional, locally finite simplicial complex X. A Dirac-element α anda dual-Dirac element β for the group Γ are elements α ∈ KKΓ(E,C) andβ ∈ KKΓ(C,E) which satisfy the following conditions

• E is a Γ-C0(X)-C∗-algebra.• The reduced crossed product E Γ is KK-equivalent to a commutative

C∗-algebra C0(Y ) where Y is an (even) Spanier-Whitehead dual of X.In particular Ktop

∗ (BΓ)−→ K∗(C0(Y ))

−→ K∗(E Γ) under thisequivalence.

• The assembly map factors as

µ : Ktop∗ (BΓ)

−→ K∗(E Γ)jr(α)−→ K∗(C∗r (Γ))

where jr is the descent transformation defined as in (5.2).• β α = 1 ∈ KKΓ(E,E)

If a Dirac element and a dual Dirac element exist the Kasparov product

γ := α β ∈ KKΓ(C,C)

is called a γ -element for the group Γ.


The element γ is an idempotent in the ring KKΓ(C,C) and jr(γ) ∈KK(C∗r (Γ), C∗r (Γ)) acts on K∗(C∗r (Γ)) as projector onto the image of theassembly map µ. Its kernel jr(1−γ) ·K∗(C∗r (Γ)) is a canonical complementof this image. If jr(γ) is equal to one, then the Baum-Connes conjectureholds for Γ and the assembly map µ is an isomorphism. In general howeverjr(γ) is different from one [Sk].

Let N (Γ) be the enveloping von Neumann algebra of C∗r (Γ) and letτ : N (Γ) → C, τ(

∑agug) := ae be the canonical faithful and positive

trace. The value of the canonical trace on the analytic index of a Γ-invariantelliptic operator D, called the von Neumann index of D, can be interpretedlike a classical index as

τ(Inda(D)) = dimN (Γ)Ker(D) − dimN (Γ)Ker(D∗)

where ordinary dimensions have to be replaced by the real valued vonNeumann dimensions. The L2-index theorem of Atiyah and Singer [At], [Si]states that the von Neumann index of D on M is equal to the ordinary indexof D on the compact manifold M

dimN (Γ)Ker(D) − dimN (Γ)Ker(D∗) = dim Ker(D) − dim Ker(D∗)

In particular, the value of the von Neumann index of a Γ-invariant ellipticoperator, which is a priori a real number, turns out to be an integer.

The Kadison-Kaplansky conjecture

Conjecture 5.4 (Kadison-Kaplansky) Let Γ be a torsion-free group and letC∗r (Γ) be its reduced group C∗-algebra, i.e. the closure in operator norm ofthe group ring CΓ acting by convolution on the Hilbert space 2(Γ). Thenthe following equivalent assertions hold:

• The spectrum of any element of C∗r (Γ) is connected.• C∗r (Γ) contains no idempotents except 0 and 1.• The value of the canonical trace on any idempotent e = e2 ∈ C∗r (Γ) is

integral: τ(e) ∈ Z.

The equivalence of the two first assertions is a consequence of holomorphicfunctional calculus and the equivalence of the two last ones follows froman elementary argument given for example in [Co].

The Kadison-Kaplansky conjecture is usually deduced as a corollary ofthe Baum-Connes conjecture. Suppose that Γ is a torsion-free group forwhich the assembly map µ : Ktop

∗ (BΓ) → K∗(C∗r (Γ)) is an isomorphismand let e = e2 ∈ C∗r (Γ) be an idempotent. Then [e] = µ(Indt(D)) =Inda(D) for some Γ-invariant elliptic operator D on a manifold M withfree and cocompact Γ-action because every element of K0(BΓ) can berealized as a topological index. Atiyah’s L2-index theorem shows then

190 M. Puschnigg

τ(e) = τ(Inda(D)) = Ind(D) ∈ Z as asserted by the third formulation ofthe Kadison-Kaplansky conjecture.

In particular, the Kadison-Kaplansky conjecture holds for all torsion-free discrete groups for which the Baum-Connes conjecture is known [HK],[La], [MY].

We interpret the integrality of the trace on idempotents as a statementabout the pairing between K-theory and local cyclic cohomology. In ourapproach both of these theories play an equally important role. The exis-tence of a γ -element for a torsion-free discrete group provides a canonicaldecomposition of the K-theory of C∗r (Γ) and of any good completion of CΓinto the direct sum of the image and the (possibly vanishing) cokernel ofthe assembly map. The local cyclic homology of good completions of CΓpossesses another canonical and natural decomposition, the homogeneousdecomposition (4.5). The central point is to show that these a priori unrelateddecompositions correspond to each other under the Chern character. Oncethis is done one argues as follows. Let a(Γ) be any good completion of CΓ.The canonical trace is a homogeneous cyclic cocycle and vanishes thereforeon the canonical complement of the assembly map in K∗(a(Γ)) because thiscomplement is mapped to the inhomogeneous part of local cyclic homologyby the Chern character. From this the integrality of the trace on idempotentsin a(Γ) follows as before by Atiyah’s L2-index theorem. This shows thatthere are no nontrivial idempotents in good completions of CΓ and if onefinds a good completion which is closed under holomorphic functional cal-culus in C∗r (Γ) the Kadison-Kaplansky conjecture for the considered groupresults.

The details of this argument are given in the next two theorems.

Theorem 5.5 Let Γ be a torsion-free discrete group with finite classifyingspace and such that HCloc∗ (1(Γ))hom H∗(Γ, HCloc∗ (C)). Let a(Γ) bea sufficiently large good completion of CΓ and let γ ∈ KKΓ(C,C) be aγ -element for Γ. Then the equivariant Chern-Connes character

cha(Γ)biv (γ) ∈ HCloc

∗ (a(Γ), a(Γ))

acts on HCloc∗ (a(Γ)) as the canonical projection onto the homogeneous partHCloc∗ (a(Γ))hom.

Proof: We proceed in several steps.

• Let a(Γ) be a sufficiently large good completion of CΓ and let cha(Γ)biv be

the equivariant Chern-Connes character (5.2).• Let γ be a γ -element for Γ and let α ∈ KKΓ(E ,C) and β ∈ KKΓ(C,E)

be the Dirac- and dual Dirac elements satisfying βα = 1 and αβ = γ .• By (5.2), (5.3) the elements cha(Γ)

biv (α) and cha(Γ)biv (β) satisfy cha(Γ)

biv (β) cha(Γ)

biv (α) = 1 and cha(Γ)biv (α) cha(Γ)

biv (β) = cha(Γ)biv (γ).


• The compatibility of the equivariant Chern-Connes character and thehomogeneous decomposition (5.2) gives rise to homomorphisms

HCloc∗ (a(Γ,E))homcha(Γ)

biv (α)−−−−−→ HCloc∗ (a(Γ))homcha(Γ)

biv (β)−−−−−→ HCloc∗ (a(Γ,E))hom⊕ ⊕ ⊕HCloc∗ (a(Γ,E))inhom

cha(Γ)biv (α)−−−−−→ HCloc∗ (a(Γ))inhom

cha(Γ)biv (β)−−−−−→ HCloc∗ (a(Γ,E))inhom

The composition of these maps equals the identity.• One finds for the homogeneous part of the local cyclic homology of the

crossed product a(Γ,E )

HCloc∗ (a(Γ,E ))hom HCloc

∗ (a(Γ,E )) HCloc∗ (E r Γ)

by (4.8)

HCloc∗ (C0(Y ))

because KK-equivalent C∗-algebras are local cyclic homology equi-valent by the multiplicativity of the bivariant Chern-Connes charac-ter [Pu2]

H∗(Γ, HCloc∗ (C))

because Y is an even Spanier-Whitehead dual of X = BΓ.• For the homogeneous part of the local cyclic homology of a(Γ) one finds

HCloc∗ (a(Γ))hom HCloc

∗ (1(Γ))hom H∗(Γ, HCloc∗ (C))

by (4.5) and by the assumption of the theorem.• In particular one obtains a commutative diagram

HCloc∗ (a(Γ,E))homcha(Γ)

biv (α)−−−−−→ HCloc∗ (a(Γ))homcha(Γ)

biv (β)−−−−−→ HCloc∗ (a(Γ,E))hom H∗(Γ, HCloc∗ (C)) H∗(Γ, HCloc∗ (C)) H∗(Γ, HCloc∗ (C))

with vertical arrows given by isomorphisms. This allows to concludethat

cha(Γ)biv (α) : HCloc

∗ (a(Γ,E ))hom → HCloc∗ (a(Γ))hom

and

cha(Γ)biv (β) : HCloc

∗ (a(Γ))hom → HCloc∗ (a(Γ,E))hom

are isomorphisms, too. If one is not willing to check the commutativityof the diagram one can deduce the assertion from the fact that all vectorspaces in sight are of the same finite dimension and from the fact thatch(β) ch(α) = 1.

192 M. Puschnigg

• Consequently cha(Γ)biv (γ)hom = cha(Γ)

biv (α)hom cha(Γ)biv (β)hom is the iden-

tity whereas cha(Γ)biv (γ)inhom = cha(Γ)

biv (α)inhom cha(Γ)biv (β)inhom vanishes

because it factors through HCloc∗ (a(Γ,E ))inhom which is zero by (4.8).In particular cha(Γ)

biv (γ) acts on HCloc∗ (a(Γ)) as the canonical projectiononto its homogeneous part.

Theorem 5.6 Let Γ be a torsion-free discrete group which satisfies thefollowing conditions:

• The classifying space BΓ has the homotopy type of a finite simplicialcomplex.

• The group ring CΓ possesses a sufficiently large good completion whichis closed under holomorphic functional calculus in C∗r (Γ).

• There exists a γ -element γ ∈ KKΓ(C,C).• HCloc∗ (1(Γ))hom H∗(Γ, HCloc∗ (C))

Then the Kadison-Kaplansky conjecture holds for Γ, i.e. C∗r (Γ) contains noidempotents except 0 and 1.

Proof: We proceed again in several steps.

• Choose a sufficiently large completion a(Γ) ofCΓ which is closed underholomorphic functional calculus in C∗r (Γ).

• Let τ be the canonical positive faithful trace on C∗r (Γ). It is concentratedon the conjugacy class of the unit. The restriction of τ to a(Γ) definestherefore a homogeneous local cyclic cocycle [τ] ∈ HC0

loc(a(Γ))hom .• For the pairing between local cyclic homology and cohomology we

deduce

〈[τ] , ch( ja(1− γ) · K∗(a(Γ)))〉 = ⟨[τ] , cha(Γ)biv (1− γ) · ch(K∗(a(Γ)))

⟩by (5.2)

= 〈πhom[τ] , πinhom ch(K∗(a(Γ)))〉 = 0

by the previous theorem. In particular the canonical trace vanishes onthe image under the Chern character of

i∗ ja(1− γ) · K∗(a(Γ)) = jr(1− γ) · (i∗K∗(a(Γ))

= jr(1− γ) · K∗(C∗r (Γ)

where i : a(Γ) → C∗r (Γ) is the inclusion because a(Γ) is closed underholomorphic functional calculus in C∗r (Γ).

• Let finally e = e2 be an idempotent in C∗r (Γ) and let [e] be its class inK-theory. Then

τ(e) = 〈[τ], ch( jr(γ) · [e])〉 + 〈[τ], ch( jr(1− γ) · [e])〉= 〈[τ], ch( jr(γ) · [e])〉 ∈ 〈[τ], ch(Im(µ))〉 ⊂ Z

by Atiyah’s L2-index theorem as explained at the beginning of thissection. This proves the Kadison-Kaplansky conjecture for Γ.


Our main application of this theorem is

Theorem 5.7 Let Γ be a torsion-free word-hyperbolic group. Then theKadison-Kaplansky conjecture holds for Γ, i.e. C∗r (Γ) contains no idempo-tents except 0 and 1.

Proof: It has to be shown that hyperbolic groups verify the conditions ofthe previous theorem.

• A torsion-free word-hyperbolic group acts freely on its Rips complexesPd(Γ) which are contractible for d 0 (2.4), (2.5). The geometricrealization |Pd(Γ)/Γ| of the quotient of Pd(Γ) by the Γ-action is a finitesimplicial complex which has the homotopy type of the classifying spaceBΓ.

• The results of Jolissaint [Jol] show that the Jolissaint algebra A(Γ) ofa word-hyperbolic group is a sufficiently large completion of the groupring which is closed under holomorphic functional calculus in C∗r (Γ).

• A γ -element for bolic and in particular word-hyperbolic groups has beenconstructed by Kasparov and Skandalis in [KS].

• The homogeneous part of the local cyclic homology of the group Ba-nach algebra 1(Γ) of a word-hyperbolic group has been calculated in(3.11) and coincides with the group homology of Γ with Z/2Z-periodiccomplex coefficients.

Remark 5.8 By the results of [Ka1] and [Pu3] the previous theorem (5.6)applies also to fundamental groups Γ of compact Riemannian manifolds ofnonpositive sectional curvature for which the group ring possesses a suffi-ciently large good completion that is closed under holomorphic functionalcalculus in C∗r (Γ). Therefore the Kadison-Kaplansky conjecture holds alsofor this class of groups. This result was previously known however byLafforgue’s proof of the Baum-Connes conjecture [La] for such groups.

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The Kadison-Kaplansky conjecture for word-hyperbolic groups · The Kadison-Kaplansky conjecture for word-hyperbolic groups Michael Puschnigg Institut de Math´ematiques de Luminy,