Siegfried Echterhoff

The structure of crossed products of irrational rotation algebras by finite subgroups of SL2(ℤ)

Abstract Let F ⊆ SL2(ℤ) be a finite subgroup (necessarily isomorphic to one of ℤ2, ℤ3, ℤ4, or ℤ6), and let F act on the irrational rotational algebra Aθ via the restriction of the canonical action of SL2(ℤ). Then the crossed product Aθ ⋊α F and the fixed point algebra are AF algebras. The same is true for the crossed product and fixed point algebra of the flip action of ℤ2 on any simple d-dimensional noncommutative torus A Θ. Along the way, we prove a number of general results which should have useful applications in other situations.

Induced Coactions of Discrete Groups on

Using the close relationship between coactions of discrete groups and Fell bundles, we introduce a procedure for inducing a C*-coaction of a quotient group G/N of a discrete group G to a C*-coaction of G itself on an induced C*-algebra. We show that induced coactions behave in many respects similarly to induced actions. In particular, as an analogue of the well known imprimitivity theorem for induced actions we prove that the crossed products of the original and the induced coactions are always Morita equivalent. We also obtain nonabelian analogues of a theorem of Olesen and Pedersen which show that there is a duality between induced coactions and twisted actions in the sense of Green. We further investigate amenability of Fell bundles corresponding to induced coactions.

C^*

We study conditions on a C ∗ -dynamical system (A,G,�) under which induction of primitive ideals (resp. irreducible representations) from stabilizers for the action of G on the primitive ideal space Prim(A) give primitive ideals (resp. irreducible representations) of the crossed product A ⋊� G. The results build on earlier results of Sauvageot (17) and others, and will correct a (possibly overly optimistic) statement of the first author in (5). In an appendix, the first author takes the opportunity to fill a gap in the proof of another result in (5).

-Algebras

In this paper we answer some questions posed by Dana Williams in [19] concerning the problem under which conditions the transformation group C*-algebra C* (G, Q) of a locally compact transformation group (G, LI) has continuous trace. One consequence will be, for compact G, that C*(G, Q2) has continuous trace if and only if the stabilizer map is continuous. We also give a complete solution to the problem if G is discrete.

INDUCING PRIMITIVE IDEALS

If a locally compact group G acts continuously via *-automorphisms on a C*-algebra A, one can form the full and reduced crossed products \(A \rtimes G \;\mathrm{and}\;A\rtimes_{r}G\) of A by G.

On transformation group *-algebras with continuous trace

We study an analogue of fibrations of topological spaces with the homotopy lifting property in the setting of C � -algebra bundles. We then derive an analogue of the Leray-Serre spectral sequence to compute the K-theory of the fibration in terms of the cohomology of the base and the K-theory of the fibres. We present many examples which show that fibrations with noncommutative fibres appear in abundance in nature. In recent years the study of the topological properties of C*-algebra bundles plays a more and more prominent role in the field of Operator algebras. The main reason for this is two-fold: on one side there are many important examples of C*-algebras which do come with a canonical bundle structure. On the other side, the study of C*-algebra bundles over a locally compact Hausdorff base space X is the natural next step in classification theory, after the far reaching results which have been obtained in the classification of simple C*-algebras. To fix notation, by a C*-algebra bundle A(X) over X we shall simply mean a C0(X)-algebra in the sense of Kasparov (see (17)): it is a C*-algebra A together with a non-degenerate ∗-homomorphism

Crossed products and the Mackey–Rieffel–Green machine

An exotic crossed product is a way of associating a C∗-algebra to each C∗-dynamical system that generalizes the well-known universal and reduced crossed products. Exotic crossed products provide natural generalizations of, and tools to study, exotic group C∗-algebras as recently considered by Brown-Guentner and others. They also form an essential part of a recent program to reformulate the Baum-Connes conjecture with coefficients so as to mollify the counterexamples caused by failures of exactness.In this paper, we survey some constructions of exotic group algebras and exotic crossed products. Summarising our earlier work, we single out a large class of crossed products—the correspondence functors—that have many properties known for the maximal and reduced crossed products: for example, they extend to categories of equivariant correspondences, and have a compatible descent morphism in K K-theory. Combined with known results on K-amenability and the Baum-Connes conjecture, this allows us to compute the K-theory of many exotic group algebras. It also gives new information about the reformulation of the Baum-Connes Conjecture mentioned above. Finally, we present some new results relating exotic crossed products for a group and its closed subgroups, and discuss connections with the reformulated Baum-Connes conjecture.

Fibrations with noncommutative fibers

In a recent paper the authors introduced universal and exotic generalized fixed-point algebras for weakly proper group actions on C*-algebras. Here we extend the notion of weakly proper actions to actions on Hilbert modules. As a result we obtain several imprimitivity theorems establishing important Morita equivalences between universal, reduced, or exotic crossed products and appropriate universal, reduced, or exotic fixed-point algebras, respectively. In particular, we obtain an exotic version of Greens imprimitivity theorem and a very general version of the symmetric imprimitivity theorem by weakly proper actions of product groups GxH. In addition, we study functorial properties of generalized fixed-point algebras for equivariant categories of C*-algebras based on correspondences.

Exotic Crossed Products

We give a complete description of the primitive ideal space of the C*-algebra associated to the ring of integers R in a number field K as considered in a recent paper by Cuntz, Deninger and Laca.

Imprimitivity theorems for weakly proper actions of locally compact groups

We examine the ideal structure of crossed products B\rtimes G where B is a continuous-trace C*-algebra and the induced action of G on the spectrum of B is proper. In particular, we are able to obtain a concrete description of the topology on the spectrum of the crossed product in the cases where either G is discrete or G is a Lie group acting smoothly on the spectrum of B.