KK-theoretic duality for proper twisted actions

Let be a smooth continuous trace algebra, with a Riemannian manifold spectrum X, equipped with a smooth action by a discrete group G such that G acts on X properly and isometrically. Then is KK-theoretically Poincaré dual to , where is the inverse of in the Brauer group of Morita equivalence classes of continuous trace algebras equipped with a group action. We deduce this from a strengthening of Kasparov’s duality theorem. As applications we obtain a version of the above Poincaré duality with X replaced by a compact G-manifold M and Poincaré dualities for twisted group algebras if the group satisfies some additional properties related to the Dirac dual-Dirac method for the Baum- Connes conjecture. This research was supported by the EU-Network Quantum Spaces and Noncommutative Geometry (Contract HPRN-CT-2002-00280) and the Deutsche Forschungsgemeinschaft (SFB 478) and by the National Science and Engineering Research Council of Canada Discovery Grant program.     

Bivariant K-theory via correspondences

We use correspondences to define a purely topological equivariant bivariant K-theory for spaces with a proper groupoid action. Our notion of correspondence differs slightly from that of Connes and Skandalis. Our construction uses no special features of equivariant K-theory. To highlight this, we construct bivariant extensions for arbitrary equivariant multiplicative cohomology theories.We formulate necessary and sufficient conditions for certain duality isomorphisms in the topological bivariant K-theory and verify these conditions in some cases, including smooth manifolds with a smooth cocompact action of a Lie group. One of these duality isomorphisms reduces bivariant K-theory to K-theory with support conditions. Since similar duality isomorphisms exist in Kasparov theory, the topological and analytic bivariant K-theories agree if there is such a duality isomorphism.     

Euler characteristics and Gysin sequences for group actions on boundaries

Let G be a locally compact group, let X be a universal proper G-space, and let be a G-equivariant compactification of X that is H-equivariantly contractible for each compact subgroup . Let . Assuming the Baum-Connes conjecture for G with coefficients and C(?X), we construct an exact sequence that computes the map on K-theory induced by the embedding . This exact sequence involves the equivariant Euler characteristic of X, which we study using an abstract notion of Poincaré duality in bivariant K-theory. As a consequence, if G is torsion-free and the Euler characteristic is non-zero, then the unit element of is a torsion element of order . Furthermore, we get a new proof of a theorem of Lück and Rosenberg concerning the class of the de Rham operator in equivariant K-homology.     

K-theory and exact sequences of partial translation algebras

In an earlier paper, the authors introduced partial translation algebras as a generalisation of group C^?${\mathrm{C}}^{?}$-algebras. Here we establish an extension of partial translation algebras, which may be viewed as an excision theorem in this context. We apply this general framework to compute the K-theory of partial translation algebras and group C^?${\mathrm{C}}^{?}$-algebras in the context of almost invariant subspaces of discrete groups. This generalises the work of Cuntz, Lance, Pimsner and Voiculescu. In particular we provide a new perspective on Pimsner's calculation of the K-theory for a graph product of groups.     

Crossed products by finite group actions with certain tracial Rokhlin property

We introduce a special tracial Rokhlin property for unital C~*-algebras. Let A be a unital tracial rank zero C~*-algebra(or tracial rank no more than one C~*-algebra). Suppose that α : G → Aut(A) is an action of a finite group G on A, which has this special tracial Rokhlin property, and suppose that A is a α-simple C~*-algebra. Then, the crossed product C~*-algebra C~*(G, A, α) has tracia rank zero(or has tracial rank no more than one). In fact,we get a more general results.     

The Baum-Connes conjecture via localisation of categories

We redefine the Baum-Connes assembly map using simplicial approximation in the equivariant Kasparov category. This new interpretation is ideal for studying functorial properties and gives analogues of the Baum-Connes assembly map for other equivariant homology theories. We extend many of the known techniques for proving the Baum-Connes conjecture to this more general setting.     

K-amenability for amalgamated free products of amenable discrete quantum groups

The basic notions and results of equivariant KK-theory concerning crossed products can be extended to the case of locally compact quantum groups. We recall these constructions and prove some useful properties of subgroups and amalgamated free products of discrete quantum groups. Using these properties and a quantum analogue of the Bass-Serre tree, we establish the K-amenability of amalgamated free products of amenable discrete quantum groups.     

Simplicity of crossed products by endomorphisms

Conditions are given for simplicity of the crossed product of a unital C^*-algebra by an endomorphism.     

Almost-invertible Toeplitz operators and K-theory

We consider the question of when a Toeplitz operator with continuous symbol on a connected compact abelian group is almost invertible, and show that this occurs precisely when the symbol is invertible and has zero topological index. The proof uses someK-theory computations.     

Bimodules in crossed products of von Neumann algebras

In this paper, we study bimodules over a von Neumann algebra M   in the context of an inclusion M⊆M_?αG$M\subseteq M{?}_{\alpha }G$, where G is a discrete group acting on a factor M by outer ?-automorphisms. We characterize the M  -bimodules X⊆M_?αG$X\subseteq M{?}_{\alpha }G$ that are closed in the Bures topology in terms of the subsets of G  . We show that this characterization also holds for w^?$w^{?}$-closed bimodules when G has the approximation property (AP  ), a class of groups that includes all amenable and weakly amenable ones. As an application, we prove a version of Mercer's extension theorem for certain w^?$w^{?}$-continuous surjective isometric maps on X.     

On topologically finite-dimensional simple <Emphasis Type="Italic">C</Emphasis>*-algebras

We show that, if a simple C*-algebra A is topologically finite-dimensional in a suitable sense, then not only K₀(A) has certain good properties, but A is even accessible to Elliott’s classification program. More precisely, we prove the following results:If A is simple, separable and unital with finite decomposition rank and real rank zero, then K₀(A) is weakly unperforated.If A has finite decomposition rank, real rank zero and the space of extremal tracial states is compact and zero-dimensional, then A has stable rank one and tracial rank zero. As a consequence, if B is another such algebra, and if A and B have isomorphic Elliott invariants and satisfy the Universal coefficients theorem, then they are isomorphic.In the case where A has finite decomposition rank and the space of extremal tracial states is compact and zero-dimensional, we also give a criterion (in terms of the ordered K₀-group) for A to have real rank zero. As a byproduct, we show that there are examples of simple, stably finite and quasidiagonal C*-algebras with infinite decomposition rank.Supported by: EU-Network Quantum Spaces - Noncommutative Geometry (Contract No. HPRN-CT-2002-00280) and Deutsche Forschungsgemeinschaft (SFB 478).     

Stable ranks of split extensions of Banach algebras

In this paper, we estimate the stable ranks of a Banach algebra in terms of the stable ranks of its quotient algebra and ideal under the assumption that the quotient map splits. As an application, several results about the Bass and the connected stable ranks of nest algebras are obtained.     

Invariant ideals of twisted crossed products

We will prove a result concerning the inclusion of non-trivial invariant ideals inside non-trivial ideals of a twisted crossed product. We will also give results concerning the primeness and simplicity of crossed products of twisted actions of locally compact groups on -algebras. Received: 25 January 2002; in final form: 22 May 2002/Published online: 2 December 2002 This work is partially supported by Hong Kong RGC Direct Grant.     

Entropy of automorphisms arising from dynamical systems through discrete groups with amenable actions

Let G⊂Aut(A) be a discrete group which is exact, that is, admits an amenable action on some compact space. Then the entropy of an automorphism of the algebra A does not change by the canonical extension to the crossed product A×G. This is shown for the topological entropy of an exact C∗-algebra A and for the dynamical entropy of an AFD von Neumann algebra A. These have applications to the case of transformations on Lebesgue spaces.     

K0 of multiplier algebras of C*-algebras with real rank zero

For a large class of -unital C ^*-algebras A with real rank zero and stable rank one, the structure of the Grothendieck group k ₀ of the multiplier algebra (A) is investigated. The ordered group K ₀( (A)) is shown to be an unperforated Riesz group, and its additive structure is completely determined, as is — in important cases — its order structure. These structures, and the attendant consequences for the ideal structure of (A), are richer than previously anticipated. In particular, it is shown that the corona algebra (A)/A can have very large stably finite quotient algebras. For example, there exist simple, separable, approximately finite-dimensional C ^*-algebras A such that the maximal stably finite quotient algebra of (A)/A has uncountably many maximal ideals modulo which a W ^*-factor of Type II₁ results. The analysis of the additive structure of K ₀( (A)) yields as a byproduct that if A is a -unital approximately finite-dimensional C ^*-algebra without nonzero unital quotient algebras, then all quasitraces on (A) are traces.This research was partially supported by a grant from the National Science Foundation.     

Spectral flow and Dixmier traces

We obtain general theorems which enable the calculation of the Dixmier trace in terms of the asymptotics of the zeta function and of the heat operator in a general semi-finite von Neumann algebra. Our results have several applications. We deduce a formula for the Chern character of an odd -summable Breuer-Fredholm module in terms of a Hochschild 1-cycle. We explain how to derive a Wodzicki residue for pseudo-differential operators along the orbits of an ergodic action on a compact space X. Finally, we give a short proof of an index theorem of Lesch for generalised Toeplitz operators.     

Quantitative K-theory for Banach algebras

Quantitative (or controlled) K-theory for $C^{?}$-algebras was used by Guoliang Yu in his work on the Novikov conjecture, and later developed more formally by Yu together with Hervé Oyono-Oyono. In this paper, we extend their work by developing a framework of quantitative K-theory for the class of algebras of bounded linear operators on subquotients (i.e., subspaces of quotients) of $L_p$ spaces. We also prove the existence of a controlled Mayer–Vietoris sequence in this framework.     

Orbifold index cobordism invariance

We prove cobordism index invariance for pseudo-differential elliptic operators on closed orbifolds with K-theoretical methods.