Local cyclic homology of group Banach algebras of "non-positively curved" discrete groups
We calculate the local cyclic homology of group Banach-algebras of discrete groups acting properly, isometrically and cocompactly on a CAT(0)-space.
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arXiv subjects
Michael Puschnigg
Publications and source records attributed to Michael Puschnigg.
Periodic cyclic homology of crossed products
We discuss the cyclic homology of crossed product algebras from the Cuntz-Quillen point of view. The periodic cyclic homology of a crossed product algebra $A\rtimes G$ is described in terms of the $G$-action on periodic cyclic bicomplexes of crossed products of $A$ by the cyclic subgroups of $G$ .
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Finitely summable $γ$-elements for word-hyperbolic groups
We present two explicit combinatorial constructions of finitely summable reduced "Gamma"-elements $γ_r\,\in\,KK(C^*_r(Γ),{\mathbb C})$ for any word-hyperbolic group $(Γ,S)$ and obtain summability bounds for them in terms of the cardinality of the generating set $S\subsetΓ$ and the hyperbolicity constant of the associated Cayley graph.
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A remark about a theorem of Skandalis
Georges Skandalis exhibited in his work on $K$-nuclearity the first class of $C^*$-algebras $A$ for which the canonical map $K_*(A\otimes_{max}A)\to K_*(A\otimes_{min}A)$ is not an isomorphism. We show that it is the injectivity that fails (even rationally) in his examples.
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Local cyclic homology of group Banach algebras I: Hyperbolic groups
We calculate the bivariant local cyclic cohomology of the Banach convolution algebra of summable functions on a word-hyperbolic group. Our result implies that the Banach algebraic assembly map in local cyclic homology is an isomorphism for such a group.
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The Baum-Connes conjecture with coefficients for word-hyperbolic groups (after Vincent Lafforgue)
Bourbaki seminar 1062, October 2012.
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The Chern-Connes character is not rationally injective
We show that the Chern-Connes character from Kasparov's bivariant K-theory to bivariant local cyclic cohomology is not always rationally injective. Counterexamples are provided by the reduced group $C^*$-algebras of word-hyperbolic groups with Kazhdan's property (T). The proof makes essential use of Skandalis' work on K-nuclearity and of Lafforgue's recent demonstration of the Baum-Connes conjecture with coefficients for word-hyperbolic groups.
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New holomorphically closed subalgebras of $C^*$-algebras of hyperbolic groups
We construct dense, unconditional subalgebras of the reduced group $C^*$-algebra of a word-hyperbolic group, which are closed under holomorphic functional calculus and possess many bounded traces. Applications to the cyclic cohomology of group $C^*$-algebras and to delocalized $L^2$-invariants of negatively curved manifolds are given.
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Finitely summable Fredholm modules over higher rank groups and lattices
We give a complete classification (up to smooth homotopy) of finitely summable Fredholm representations (Fredholm modules) over higher rank groups and lattices. Our results are a direct consequence of work of Bader, Furman, Gelander and Monod on generalisations of Kazhdan's property T for locally compact groups.
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Characters of Fredholm modules and a problem of Connes
It is shown that Connes' character formula for unbounded, theta-summable Fredholm modules represents the abstract Chern-character in K-homology. As an application, the character of a particular Fredholm module over the reduced group C*-algebra of a discrete group is calculated, which leads to a partial solution of a problem posed by A.Connes.
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