Search

For a large class of $$C^*$$ C ∗ -algebras A, we calculate the K-theory of reduced crossed products $$A^{\otimes G}\rtimes _rG$$ A ⊗ G ⋊ r G of Bernoulli shifts by groups satisfying the Baum–Connes conjecture. In particular, we give explicit formulas for finite-dimensional $$C^*$$ C ∗ -algebras, UHF-algebras, rotation algebras, and several other examples. As an application, we obtain a formula for the K-theory of reduced $$C^*$$ C ∗ -algebras of wreath products $$H\wr G$$ H ≀ G for large classes of groups H and G. Our methods use a generalization of techniques developed by the second named author together with Joachim Cuntz and Xin Li, and a trivialization theorem for finite group actions on UHF algebras developed in a companion paper by the third and fourth named authors.

This paper is motivated primarily by the question of when the maximal and reduced crossed products of a $G$-$C^*$-algebra agree (particularly inspired by results of Matsumura and Suzuki), and the relationships with various notions of amenability and injectivity. We give new connections between these notions. Key tools in this include the natural equivariant analogues of injectivity, and of Lance's weak expectation property: we also give complete characterizations of these equivariant properties, and some connections with injective envelopes in the sense of Hamana., Comment: 32 pages

The purpose of this paper is to introduce a consistent notion of universal and reduced crossed products by actions and coactions of groups on operator systems and operator spaces. In particular we shall put emphasis to reveal the full power of the universal properties of the the universal crossed products. It turns out that to make things consistent, it seems useful to perform our constructions on some bigger categories which allow the right framework for studying the universal properties and which are stable under the construction of crossed products even for non-discrete groups. In the case of operator systems, this larger category is what we call a $C^*$-operator system, i.e., a selfadjoint subspace $X$ of some $\mathcal B(H)$ which contains a $C^*$-algebra $A$ such that $AX=X=XA$. In the case of operator spaces, the larger category is given by what we call $C^*$-operator bimodules. After we introduced the respective crossed products we show that the classical Imai-Takai and Katayama duality theorems for crossed products by group (co-)actions on $C^*$-algebras extend one-to-one to our notion of crossed products by $C^*$-operator systems and $C^*$-operator bimodules., Comment: This paper is a completely rewritten and reorganized replacement of the paper "Crossed products by operator spaces" (see arXiv:1512.07776)

We study general properties of exotic crossed-product functors and characterise those which extend to functors on equivariant C * C^{*} -algebra categories based on correspondences. We show that every such functor allows the construction of a descent in K K-theory and we use this to show that all crossed products by correspondence functors of K-amenable groups are K K-equivalent. We also show that for second countable groups the minimal exact Morita compatible crossed-product functor used in the new formulation of the Baum–Connes conjecture by Baum, Guentner and Willett ([‘Expanders, exact crossed products, and the Baum–Connes conjecture’, preprint 2013]) extends to correspondences when restricted to separable G- C * C^{*} -algebras. It therefore allows a descent in K K-theory for separable systems.

A crossed product functor is said to be injective if it takes injective morphisms to injective morphisms. In this paper we show that every locally compact group $G$ admits a maximal injective crossed product $A\mapsto A\rtimes_{\inj}G$. Moreover, we give an explicit construction of this functor that depends only on the maximal crossed product and the existence of $G$-injective $C^*$-algebras; this is a sort of a `dual' result to the construction of the minimal exact crossed product functor, the latter having been studied for its relationship to the Baum-Connes conjecture. It turns out that $\rtimes_\inj$ has interesting connections to exactness, the local lifting property, amenable traces, and the weak expectation property., 18 pages

Given a locally compact group $G$, we study the smallest exact crossed-product functor $(A,G,\alpha)\mapsto A\rtimes_{\mathcal E} G$ on the category of $G$-$C^*$-dynamical systems. As an outcome, we show that the smallest exact crossed-product functor is automatically Morita compatible, and hence coincides with the functor $\rtimes_{\mathcal{E}}$ as introduced by Baum, Guentner, and Willett in their reformulation of the Baum-Connes conjecture (see [2]). We show that the corresponding group algebra $C_{\mathcal{E}}^*(G)$ always coincides with the reduced group algebra, thus showing that the new formulation of the Baum-Connes conjecture coincides with the classical one in the case of trivial coefficients. Erratum: After publication of this manuscript, some gaps have unfortunately been found affecting some parts of the paper. We therefore included an appendix with an erratum at the end of this paper explaining the mistakes and keeping the original published version unchanged., Comment: 37 pages

ALCIDES BUSS AND SIEGFRIED ECHTERHOFFAbstract. Using the theory of weakly proper actions of locally compactgroups recently developed by the authors, we give a unified proof of both re-duced and maximal versions of Mansfield’s Imprimitivity Theorem and obtaina general version of Landstad’s Duality Theorem for twisted group coactions.As one application, we obtain the stabilization trick for arbitrary twisted coac-tions, showing that every twisted coaction is Morita equivalent an inflatedcoaction.

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.

The extension of K-theory from topological spaces to operator algebras provides the most powerful tool for the study of C *-algebras. On one side there now exist far reaching classification results in which certain classes of C *-algebras can be classified by their K-theoretic data. This started with the early work of Elliott [Ell76] on the classification of AF-algebras – inductive limits of finite-dimensional C *-algebras. It went on with the classification of simple, separable, nuclear, purely infinite C *-algebras by Kirchberg and Phillips [KP00,Phi00]. Presently, due to the work of many authors (e.g., see [Win16] for a survey on the most recent developments) the classification program covers a very large class of nuclear algebras.

Let P be a semigroup that admits an embedding into a group G. Assume that the embedding satisfies a certain Toeplitz condition and that the Baum-Connes conjecture holds for G. We prove a formula describing the K- theory of the reduced crossed product A \rtimes{\alpha},r P by any automorphic action of P. This formula is obtained as a consequence of a result on the K-theory of crossed products for special actions of G on totally disconnected spaces. We apply our result to various examples including left Ore semigroups and quasi-lattice ordered semigroups. We also use the results to show that for certain semigroups P, including the ax + b-semigroup for a Dedekind domain R, the K-theory of the left and right regular semigroup C*-algebras of P coincide, although the structure of these algebras can be very different.

The purpose of this paper is to give a complete description of the primitive ideal space of the C*-algebra [R] associated to the ring of integers R in a number field K in the recent paper [5]. As explained in [5], [R] can be realized as the Toeplitz C*-algebra of the affine semigroup R ⋊ R× over R and as a full corner of a crossed product C0() ⋊ K ⋊ K*, where is a certain adelic space. Therefore Prim([R]) is homeomorphic to the primitive ideal space of this crossed product. Using a recent result of Sierakowski together with the fact that every quasi-orbit for the action of K ⋊ K* on contains at least one point with trivial stabilizer we show that Prim([R]) is homeomorphic to the quasi-orbit space for the action of K ⋊ K* on , which in turn may be identified with the power set of the set of prime ideals of R equipped with the power-cofinite topology.

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.

Using Poincar\'e duality in K-theory, we state and prove a Lefschetz fixed point formula for endomorphisms of cross product C*-algebras $C_0(X)\cross G$ coming from covariant pairs. Here $G$ is assumed countable, $X$ a manifold, and $X\cross G$ cocompact and proper. The formula in question expresses the graded trace of the map on rationalized K-theory of $C_0(X)\cross G$ induced by the endomorphism, \emph{i.e.} the Lefschetz number, in terms of fixed orbits and representation-theoretic data connected with certain isotropy subgroups of the isotropy group at that point. The technique is to use noncommutative Poinca\'e duality and the formal Lefschetz lemma of the second author., Comment: 26 pages

Let the discrete group G act properly and isometrically on the Riemannian manifold X. Let C_0(X, \delta) be the section algebra of a smooth locally trivial G-equivariant bundle of elementary C*-algebras representing an element \delta of the Brauer group Br_G(X). Then C_0(X,\delta^{-1}) x G is KK-theoretically Poincare dual to (C_0(X,\delta)\otimes_{C_0(X)} C_\tau(X)) xG, where \delta^{-1} is the inverse of \delta in the Brauer group. We deduce this from a strengthening of Kasparov's duality theorem RKK^G(X; A,B) \cong KK^G(C_\tau(X)\otimes A, B). As applications we also obtain a version of the above Poincare duality with X replaced by a compact G-manifold M and for twisted group algebras C*(G,\omega) if G satisfies some additional properties related to the Dirac-dual Dirac method for the Baum-Connes conjecture., Comment: 20 pages, organizational and editorial changes, minor additions

Let G be a locally compact group with cocompact connected component. We prove that the assembly map from the topological K-theory of G to the K-theory of the reduced C*-algebra of G is an isomorphism. The same is shown for the groups of k-rational points of any linear algebraic group over a local field k of characteristic zero.

We examine the structure of central twisted transformation group C � -algebras C0(X) ⋊id,u G, and apply our results to the group C � -algebras of central group extensions. Our methods require that we study Moore's coho- mology group H 2 G, C(X, T) � , and, in particular, we prove an inflation result for pointwise trivial cocyles which may be of use elsewhere. The study of twisted transformation groupC ∗ -algebras and Moore's cohomology group H 2 G,C(X,T) � arise naturally in the study of the group C ∗ -algebras of group extensions and have far reaching applications in operator algebras. Packer's survey articles (13), and especially (12), give an excellent historical context as well as providing additional references and the basic definitions for what follows. In this paper we examine the structure of central twisted transformation group C ∗ -algebras C0(X) ⋊id,u G where G is a second countable locally com- pact group acting trivially on a second countable locally compact space X and u ∈ Z 2 G,C(X,T) � is a 2-cocycle on G taking values in the center C(X,T) of the multiplier algebra of C0(X). A primary motivation is the study of the group C ∗ -algebra C ∗ (L) of a central group extension

In the late 1980's Marc Rieffel introduced a notion of properness for actions of locally compact groups on C*-algebras which, among other things, allows the construction of generalised fixed-point algebras for such actions. In this paper we give a simple characterisation of Rieffel proper actions and use this to obtain several (counter) examples for the theory. In particular, we provide examples of Rieffel proper actions $\alpha:G\to\mathrm{Aut}(A)$ for which properness is not induced by a nondegenerate equivariant *-homomorphism $\phi:C_0(X)\to \mathcal{M}(A)$ for any proper $G$-space $X$. Other examples, based on earlier work of Meyer, show that a given action might carry different structures for Rieffel properness with different generalised fixed-point algebras., Comment: 19 pages

In this chapter we prove the Stone-von Neumann Theorem, which gives a full characterization of the unitary dual of the Heisenberg group \({\cal H}\). We then apply the trace formula to describe the spectral decomposition of \({L^2}(\Lambda \backslash H)\), where π is the standard integer lattice in \({\cal H}\).

The majority of the examples of topological groups in this book given so far, are locally euclidean, meaning that the groups are locally homeomorphic to \({\mathbb R}^n.\) In this chapter the reader will see some examples which are not of this type. These examples, the p-adic numbers and the adeles, resp. ideles, are not only interesting as examples of this theory, but they also carry great importance for other areas of mathematics, in particular number theory.

In this chapter we will apply the duality theorem for proving structure theorems for LCA groups. As main result we will show that all such groups are isomorphic to groups of the form \({\mathbb R}^n\times H\) for some \(n\in{\mathbb N}_0\), such that H is a locally compact abelian group that contains an open compact subgroup K. This theorem will imply better structure theorems if more information on the group is available. For instance it will follow that every compactly generated LCA group is isomorphic to a group of the form \({\mathbb R}^n\times{\mathbb Z}^m\times K\) for some compact group K and some \(n,m\in{\mathbb N}_0\), and every compactly generated locally euclidean group is isomorphic to one of the form \({\mathbb R}^n\times{\mathbb Z}^m\times{\mathbb T}^l\times F\), for some finite group F and some nonnegative integers \(n,m\) and l.

In this chapter we are mainly interested in the study of abelian locally compact groups A, their dual groups \(\hat{A}\) together with various associated group algebras. Using the Gelfand-Naimark Theorem as a tool, we shall then give a proof of the Plancherel Theorem, which asserts that the Fourier transform extends to a unitary equivalence of the Hilbert spaces \(L^2(A)\) and \(L^2(\widehat A)\). We also prove the Pontryagin Duality Theorem that gives a canonical isomorphism between A and its bidual \({\hat{\hat{A}}}\).

In this chapter, we will apply the results of Chap. 2 on \(C^*\)-algebras to operators on Hilbert spaces. In particular, we will discuss the continuous functional calculus for normal bounded operators on Hilbert space, which turns out to be a powerful tool.

In this chapter we introduce the Selberg trace formula, which is a natural generalization of the Poisson summation formula to non-abelian groups. Applications of the trace formula will be given in the next two chapters.

The group \({\rm SL}_2({\mathbb R})\) is the simplest case of a so called reductive Lie group. Harmonic analysis on these groups turns out to be more complex then the previous cases of abelian, compact, or nilpotent groups. On the other hand, the applications are more rewarding. For example, via the theory of automorphic forms, in particular the Langlands program, harmonic analysis on reductive groups has become vital for number theory. In this chapter we prove an explicit Plancherel Theorem for functions in the Hecke algebra of the group \(G={\rm SL}_2({\mathbb R})\). We apply the trace formula to a uniform lattice and as an application derive the analytic continuation of the Selberg zeta function.

Resume Soit G un groupe localement compact. Dans une premiere partie, nous montrons comment un espace topologique muni d'une action propre de G peut etre localement vu comme induit a partir d'un espace muni d'une action d'un sous-groupe compact de G . Pour l'application de Baum–Connes, ceci montre que dans la definition de la K -theorie topologique d'un groupe, rien n'est change si la notion classique d'action propre de G remplace la notion introduite par Baumes, Connes et Higson (1994). Dans une deuxieme partie, nous montrons que l'application de Baum–Connes pour G et a coefficients dans une algebre propre est un isomorphisme.

The symmetric imprimitivity theorem provides a Morita equivalence between two crossed products of induced C*-algebras and includes as special cases many other important Morita equivalences such as Green's imprimitivity theorem. We show that the symmetric imprimitivity theorem is compatible with various inflated actions and coactions on the crossed products.

Using the close relationship between coactions of discrete groups and Fell bundles, we introduce a procedure for inducing a C*-coaction δ: D → D ⊗C*(G/N) of a quotient group G/N of a discrete group G to a C*-coaction Ind δ: Ind D → D ⊗C*(G) of G. 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 Ind D ×IndδG and D ×δG/N 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.

Suppose thatGhas a representation groupH, thatGab≔G/ [G, G] is compactly generated, and thatAis aC*-algebra for which the complete regularization of Prim(A) is a locally compact Hausdorff spaceX. In a previous article, we showed that there is a bijectionα↦(Zα, fα) between the collection of exterior equivalence classes of locally inner actionsα: G→Aut(A), and the collection of principalGab-bundlesZαtogether with continuous functionsfα: X→H2(G, T ). In this paper, we compute the crossed productsA⋊αGin terms of the dataZα,fα, andC*(H).

Given a C*-dynamical system (A,G,\alpha), we say that A is a weakly proper (X\rtimes G)-algebra if there exists a proper G-space X together with a nondegenerate G-equivariant *-homomorphism \phi:C_0(X)->M(A). Weakly proper G-algebras form a large subclass of the class of proper G-algebras in the sense of Rieffel. In this paper we show that weakly proper (X\rtimes G)-algebras allow the construction of full fixed-point algebras A^G corresponding to the full crossed product A\rtimes_{\alpha}G, thus solving, in this setting, a problem stated by Rieffel in his 1988's original article on proper actions. As an application we obtain a general Landstad duality result for arbitrary coactions together with a new and functorial construction of maximalizations of coactions. The same methods also allow the construction of exotic generalized fixed-point algebras associated to crossed-product norms lying between the reduced and universal ones. Using these, we give complete answers to some questions on duality theory for exotic crossed products recently raised by Kaliszewski, Landstad and Quigg., Comment: 32 pages, revised version

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 Green's 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., Comment: The article has been completely revised. In the second version, the main part of the paper on imprimitivity theorems has been re-written and generalized to exotic crossed products. The third version essencially only contains minor changes and improvements in the text. The article has now 43 pages. Version accepted for publications in Erg. Theory and Dyn. Systems