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Über dieses Buch

This book contains a collection of articles provided by the participants of the SFB-workshop on C*-algebras, March 8 - March 12, 1999 which was held at the Sonderforschungsbereich "Geometrische Strukturen in der reinen Mathematik" of the University of Münster, Germany. The aim of the workshop was to bring together leading experts in the theory of C* -algebras with promising young researchers in the field, and to provide a stimulating atmosphere for discussions and interactions between the participants. There were 19 one-hour lectures on various topics like - classification of nuclear C* -algebras, - general K-theory for C* -algebras, - exact C* -algebras and exact groups, - C*-algebras associated to (infinite) matrices and C*-correspondences, - noncommutative prob ability theory, - deformation quantization, - group C* -algebras and the Baum-Connes conjecture, giving a broad overview of the latest developments in the field, and serving as a basis for discussions. We, the organizers of the workshop, were greatly pleased with the excellence of the lectures and so were led to the idea of publishing the proceedings of the conference. There are basically two kinds of contributions. On one side there are several articles giving surveys and overviews on new developments and im­ portant results of the theory, on the other side one finds original articles with interesting new results.



Some Properties of C*-Algebras Associated to Discrete Linear Groups

Let Γ be a discrete (countable) group. There are two distinguished C*-algebras one may associate to Γ :
  • the reduced C*-algebra C* r (Γ) which is the norm closure of the linear span of {(λ (γ) : γ ∈ Γ}, where λ is the left regular representation of Γ on ℓ2 (Γ), defined by
    $$lambda (\gamma)f(x) = f({\gamma ^{ - 1}}x)\forall \gamma ,{\text{x}} \in \Gamma ,f \in {\ell ^2}(\Gamma);$$
  • the maximal (or full) C* -algebra C*(Γ) which is the completion of the group algebra ℂΓ for the C*-norm
    $$parallel \sum {{c_\gamma }\gamma {\parallel _{{\text{max}}}} = {\text{sup}}\left\{ \parallel \right.} \sum {{c_\gamma }\pi (\gamma )\parallel :\pi } {\text{ is a}}*{\text{-representation of }}\mathbb{C}\left. \Gamma \right\}$$
M. B. Bekka, N. Louvet

Generalized Inductive Limits and Quasidiagonality

This survey article describes the connection between the theory of generalized inductive limits of finite-dimensional C*-algebras and quasidiagonality. Connections with the classification problem for separable nuclear C*-algebras are also discussed.
Bruce Blackadar, Eberhard Kirchberg

Approximate Unitary Equivalence and the Topology of Ext (A, B)

Let A, B be unital C*-algebras and assume that A is separable and quasidiagonal relative to B. Let ϕ, ψ : AB be unital *-homomorphisms. If A is nuclear and satisfies the UCT, we prove that ϕ is approximately stably unitarily equivalent to ψ if and only if ϕ* = ψ* : K * (A, ℤ/n) → K *(B, ℤ/n) for all n ≥ 0. We give a new proof of a result of [DE2] which states that if A is separable and quasidiagonal relative to B and if (ϕ, ψ : AB have the same KK-class, then (ϕ is approximately stably unitarily equivalent to ψ. For nuclear separable C*-algebras A, we give a KK-theoretical description of the closure of zero in Ext(A, B).
Marius Dadarlat

Free Products of Exact Groups

It has recently been proved that the class of unital exact C*-algebras is closed under taking reduced amalgamated free products. Here the proof is presented of a special case: that the class of exact discrete groups is closed under taking free products (with amalgamation over the identity element). The proof of this special case is considerably simpler than in full generality.
Kenneth J. Dykema

Random Matrices and Non-Exact C*-Algebras

In the paper [HT2], we gave new proofs based on random matrix methods of the following two results:
Any unital exact stably finite C*-algebra has a tracial state.
If A is a unital exact C*-algebra, then any state on K 0(A) comes from a tracial state on A.
U. Haagerup, S. Thorbjørnsen

Das nicht-kommutative Michael-Auswahlprinzip und die Klassifikation nicht-einfacher Algebren

We outline the proofs of the following results (1)–(4):
Suppose that A and B are separable nuclear stable C*-algebras and that γ is a homeomorphism from Prim(A) onto Prim(B), then there is an isomorphism ϕ from AO 2 onto BO 2 such that, for J ∈ Prim(A),
$$varphi (J \otimes {O_2}) = \gamma (J) \otimes {O_2}$$
Eberhard Kirchberg

C*-Algebraic Deformation Quantization of Closed Riemann Surfaces

We study an explicit construction of C*-algebraic deformation quantization of closed Riemann surfaces.
Toshikazu Natsume

Index of Γ-Equivariant Toeplitz Operators

Let Γ be a discrete subgroup of PSL(2,ℝ) of infinite covolume with infinite conjugacy classes. H t denotes the Hilbert space consisting of analytic functions in \({L^2}(\mathbb{D},{({\text{Im }}z)^{t-2}}{\text{d}}\bar z{\text{d}}z)\) and, for t > 1, π t denotes the corresponding projective unitary representation of PSL(2, ℝ) on this Hilbert space. Let A t be the II∞ factor given by the commutant of π t(Γ) in B(H t). Let F denote a fundamental domain for Γ in D. We assume that t > 5 and give \(partial M = \partial \mathbb{D} \cap \bar F\) the topology of disjoint union of its connected components.
Ryszard Nest, Florin Radulescu

Twisted Actions and Obstructions in Group Cohomology

This article is intended to answer the question “Why do you guys always want to twist everything?” We review the various ways in which twists, twisted actions and twisted crossed products arise, and then discuss some cohomological obstructions to the existence and triviality of twisted actions.
Iain Raeburn, Aidan Sims, Dana P. Williams

Boundary Actions for Affine Buildings and Higher Rank Cuntz-Krieger Algebras

Let Γ be a group of type rotating automorphisms of an affine building B of type à 2. If Γ acts freely on the vertices of B with finitely many orbits, and if Ω is the (maximal) boundary of B then C(Ω) Γ is a p.i.s.u.n. C*-algebra. This algebra has a structure theory analogous to that of a simple Cuntz-Krieger algebra and is the motivation for a theory of higher rank Cuntz-Krieger algebras, which has been developed by T. Steger and G. Robertson. The K-theory of these algebras can be computed explicitly in the rank two case. For the rank two examples of the form C(Ω) Γ which arise from boundary actions on à 2 buildings, the two K-groups coincide.
Guyan Robertson

Crossed Products by C*-Correspondences and Cuntz-Pimsner Algebras

We introduce Cuntz-Pimsner algebras from the point of view of crossed products by C*-correspondences. Strong emphasis is put on the discussion of examples. In particular, we show that the Cuntz-Krieger algebras for infinite matrices recently introduced by Exel and Laca are Cuntz-Pimsner algebras, leading to a much simplified computation of their K-theory, as well as a characterization of simplicity in the unital case.
Jürgen Schweizer

The Baum-Connes Conjecture for Groupoids

This survey paper is a self-contained overview on the Baum-Connes conjecture for locally compact groupoids.
Jean-Louis Tu

C*-Exact Groups

We report on joint work with E. Kirchberg on C*-exact groups, which will appear in [KW1] and [KW2], and discuss some of the open problems in the area.
Simon Wassermann

Some Free Ordered C*-Modules

A free normed module XF over the (complex) algebra F of finite dimensional operators on a separable Hilbert space H 0 is called an operator space if it is isometrical isomorphic to a submodule of L(H 1) ⊗min F, where ⊗min denotes the minimal (or spatial) tensor product. One might consider operator spaces as ‘non-commutative’ normed spaces because, formally, the scalar field has been replaced by F
Wend Werner

Quasi-free Automorphisms of Cuntz-Krieger-Pimsner Algebras

We consider quasi-free automorphisms of C*-algebras O E generated by Hilbert bimodules E extending the classical notion for O n . We obtain results about pure infiniteness of O E and simplicity of crossed products by groups of quasi-free automorphisms. Using that we find some new examples of stably projectionless simple C*-algebras as crossed products of purely infinite algebras.
Joachim Zacharias
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