arXiv subjects

Alcides Buss

Publications and source records attributed to Alcides Buss.

53 records · Page 3Linked to original sources

arXiv2015

Maximality of dual coactions on sectional C*-algebras of Fell bundles and applications

In this paper we give a simple proof of the maximality of dual coactions on full cross-sectional C*-algebras of Fell bundles over locally compact groups. As applications we extend certain exotic crossed-product functors in the sense of Baum, Guentner and Willett to the category of Fell bundles and the category of partial actions and we obtain results about the K-theory of (exotic) cross-sectional algebras of Fell-bundles over K-amenable groups. As a bonus, we give a characterisation of maximal coactions of discrete groups in terms of maximal tensor products.

math.OA↗

arXiv2015

Exotic crossed products and the Baum-Connes conjecture

We study general properties of exotic crossed-product functors and characterise those which extend to functors on equivariant C*-algebra categories based on correspondences. We show that every such functor allows the construction of a descent in KK-theory and we use this to show that all crossed products by correspondence functors of K-amenable groups are KK-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 extends to correspondences when restricted to separable G-C*-algebras. It therefore allows a descent in KK-theory for separable systems.

math.OA↗

arXiv2015

Crossed products for actions of crossed modules on C*-algebras

We decompose the crossed product functor for actions of crossed modules of locally compact groups on C*-algebras into more elementary constructions: taking crossed products by group actions and fibres in C*-algebras over topological spaces. For this, we extend the theory of partial crossed products from groups to crossed modules; extend Takesaki-Takai duality to Abelian crossed modules; show that equivalent crossed modules have equivalent categories of actions on C*-algebras; and show that certain crossed modules are automatically equivalent to Abelian crossed modules.

math.OA↗

arXiv2014

Rieffel proper actions

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 $α:G\to\mathrm{Aut}(A)$ for which properness is not induced by a nondegenerate equivariant *-homomorphism $ϕ:C_0(X)\to \mathcal{M}(A)$ 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.

math.OA↗

arXiv2014

Universal and exotic generalized fixed-point algebras for weakly proper actions and duality

Given a C*-dynamical system (A,G,α), 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 ϕ: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_αG, 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.

math.OA↗

arXiv2014

Weakly proper group actions, Mansfield's imprimitivity and twisted Landstad duality

Using the theory of weakly proper actions of locally compact groups recently developed by the authors, we give a unified proof of both reduced and maximal versions of Mansfield's Imprimitivity Theorem and obtain a general version of Landstad's Duality Theorem for twisted group coactions. As one application, we obtain the stabilization trick for arbitrary twisted coactions, showing that every twisted coaction is Morita equivalent to an inflated coaction.

math.OA↗

arXiv2014

Imprimitivity theorems for weakly proper actions of locally compact groups

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.

math.OA↗

arXiv2013

Inverse semigroup expansions and their actions on C*-algebras

In this work, we give a presentation of the prefix expansion Pr(G) of an inverse semigroup G as recently introduced by Lawson, Margolis and Steinberg which is similar to the universal inverse semigroup defined by the second named author in case G is a group. The inverse semigroup Pr(G) classifies the partial actions of G on spaces. We extend this result and prove that Fell bundles over G correspond bijectively to saturated Fell bundles over Pr(G). In particular, this shows that twisted partial actions of G (on C*-algebras) correspond to twisted (global) actions of Pr(G). Furthermore, we show that this correspondence preserves C*-algebras crossed products.

math.OA↗

arXiv2013

Generalized fixed point algebras for coactions of locally compact quantum groups

We extend the construction of generalized fixed point algebras to the setting of locally compact quantum groups - in the sense of Kustermans and Vaes - following the treatment of Marc Rieffel, Ruy Exel and Ralf Meyer in the group case. We mainly follow Meyer's approach analyzing the constructions in the realm of equivariant Hilbert modules. We generalize the notion of continuous square-integrability, which is exactly what one needs in order to define generalized fixed point algebras. As in the group case, we prove that there is a correspondence between continuously square-integrable Hilbert modules over an equivariant C*-algebra B and Hilbert modules over the reduced crossed product of B by the underlying quantum group. The generalized fixed point algebra always appears as the algebra of compact operators of the associated Hilbert module over the reduced crossed product.

math.OA↗

arXiv2010

Twisted actions and regular Fell bundles over inverse semigroups

We introduce a new notion of twisted actions of inverse semigroups and show that they correspond bijectively to certain regular Fell bundles over inverse semigroups, yielding in this way a structure classification of such bundles. These include as special cases all the stable Fell bundles. Our definition of twisted actions properly generalizes a previous one introduced by Sieben and corresponds to Busby-Smith twisted actions in the group case. As an application we describe twisted etale groupoid C*-algebras in terms of crossed products by twisted actions of inverse semigroups and show that Sieben's twisted actions essentially correspond to twisted etale groupoids with topologically trivial twists.

math.OA↗

arXiv2009

Non-Hausdorff Symmetries of C*-algebras

Symmetry groups or groupoids of C*-algebras associated to non-Hausdorff spaces are often non-Hausdorff as well. We describe such symmetries using crossed modules of groupoids. We define actions of crossed modules on C*-algebras and crossed products for such actions, and justify these definitions with some basic general results and examples.

math.OA↗

arXiv2009

Fell bundles over inverse semigroups and twisted etale groupoids

Given a saturated Fell bundle A over an inverse semigroup S which is semi-abelian in the sense that the fibers over the idempotents of S are commutative, we construct a twisted etale groupoid L such that A can be recovered from L in a canonical way. As an application we recover most of Renault's recent result on the classification of Cartan subalgebras of C*-algebras through twisted etale groupoids.

math.OA↗

arXiv2009

Integrability of dual coactions on Fell bundle C*-algebras

We study integrability for coactions of locally compact groups. For abelian groups, this corresponds to integrability of the associated action of the Pontrjagin dual group. The theory of integrable group actions has been previously studied by Ruy Exel, Ralf Meyer and Marc Rieffel. Our goal is to study the close relationship between integrable group coactions and Fell bundles. As a main result, we prove that dual coactions on C*-algebras of Fell bundles are integrable, generalizing results by Ruy Exel for abelian groups.

math.OA↗

arXiv2009

A higher category approach to twisted actions on C*-algebras

C*-algebras form a 2-category with \Star{}homomorphisms or correspondences as morphisms and unitary intertwiners as 2-morphisms. We use this structure to define weak actions of 2-categories, weakly equivariant maps between weak actions, and modifications between weakly equivariant maps. In the group case, we identify the resulting notions with known ones, including Busby-Smith twisted actions and equivalence of such actions, covariant representations, and saturated Fell bundles. For 2-groups, weak actions combine twists in the sense of Green and Busby-Smith. The Packer-Raeburn Stabilisation Trick implies that all Busby-Smith twisted group actions of locally compact groups are Morita equivalent to classical group actions. We generalise this to actions of strict 2-groupoids.

math.OA↗

arXiv2009

A generalized Fourier inversion theorem

In this work we define operator-valued Fourier transforms for suitable integrable elements with respect to the Plancherel weight of a (not necessarily Abelian) locally compact group. Our main result is a generalized version of the Fourier inversion Theorem for strictly-unconditionally integrable Fourier transforms. Our results generalize and improve those previously obtained by Ruy Exel in the case of Abelian groups.

math.FA↗

arXiv2009

Square-integrable coactions of locally compact quantum groups

We define and study square-integrable coactions of locally compact quantum groups on Hilbert modules, generalising previous work for group actions. As special cases, we consider square-integrable Hilbert space corepresentations and integrable coactions on C*-algebras. Our main result is an equivariant generalisation of Kasparov's Stabilisation Theorem.

math.OA↗

arXiv2007

Continuous spectral decompositions of Abelian group actions on C*-algebras

Let G be a locally compact Abelian group. Following Ruy Exel, we view Fell bundles over the Pontrjagin dual of G as continuous spectral decompositions of G-actions on C*-algebras. We classify such spectral decompositions using certain dense subspaces related to Marc Rieffel's theory of square-integrability. There is a unique continuous spectral decomposition if the group acts properly on the primitive ideal space of the C*-algebra. But there are also examples of group actions without or with several inequivalent spectral decompositions.

math.OA↗