scholarly journals The Non-selfadjoint Approach to the Hao–Ng Isomorphism

2019 ◽  
Author(s):  
Elias G Katsoulis ◽  
Christopher Ramsey
Keyword(s):  
Discrete Group ◽  
Operator Algebras ◽  
Finite Graph ◽  
Isomorphism Problem ◽  
Crossed Product ◽  
Compact Groups ◽  
Crossed Products ◽  
Injective Envelope ◽  
Tensor Algebras ◽  
Dilation Theory

Abstract In an earlier work, the authors proposed a non-selfadjoint approach to the Hao–Ng isomorphism problem for the full crossed product, depending on the validity of two conjectures stated in the broader context of crossed products for operator algebras. By work of Harris and Kim, we now know that these conjectures in the generality stated may not always be valid. In this paper we show that in the context of hyperrigid tensor algebras of $\mathrm{C}^*$-correspondences, each one of these conjectures is equivalent to the Hao–Ng problem. This is accomplished by studying the representation theory of non-selfadjoint crossed products of C$^*$-correspondence dynamical systems; in particular we show that there is an appropriate dilation theory. A large class of tensor algebras of $\mathrm{C}^*$-correspondences, including all regular ones, are shown to be hyperrigid. Using Hamana’s injective envelope theory, we extend earlier results from the discrete group case to arbitrary locally compact groups; this includes a resolution of the Hao–Ng isomorphism for the reduced crossed product and all hyperrigid $\mathrm{C}^*$-correspondences. A culmination of these results is the resolution of the Hao–Ng isomorphism problem for the full crossed product and all row-finite graph correspondences; this extends a recent result of Bedos, Kaliszewski, Quigg, and Spielberg.

Ergodic Actions of Compact Groups on Operator Algebras II: Classification of Full Multiplicity Ergodic Actions

1988 ◽  
Vol 40 (6) ◽  
pp. 1482-1527 ◽  
Author(s):  
Antony Wassermann
Keyword(s):  
Von Neumann Algebras ◽  
Operator Algebras ◽  
Crossed Product ◽  
Ergodic Action ◽  
Compact Groups ◽  
Type I ◽  
List Type ◽  
Von Neumann ◽  
C Algebra ◽  
Set Up

In the first paper of this series [17], we set up some general machinery for studying ergodic actions of compact groups on von Neumann algebras, namely, those actions for which . In particular we obtained a characterisation of the full multiplicity ergodic actions:THEOREM A. If α is an ergodic action of G on , then the following conditions are equivalent:(1) Each spectral subspace has multiplicity dim π for π in .(2) Each π in admits a unitary eigenmatrix in .(3) The W* crossed product is a (Type I) factor.(4) The C* crossed product of the C* algebra of norm continuity is isomorphic to the algebra of compact operators on a Hilbert space.

scholarly journals Equivariant KK-Theory and the Continuous Rokhlin Property

2020 ◽  
Author(s):  
Eusebio Gardella
Keyword(s):  
Fixed Point ◽  
Crossed Product ◽  
Compact Groups ◽  
Crossed Products ◽  
Circle Actions ◽  
Technical Result ◽  
C Algebras ◽  
Equivariant Version ◽  
Universal Coefficient Theorem

Abstract We introduce and study the continuous Rokhlin property for actions of compact groups on $C^*$-algebras. An important technical result is a characterization of the continuous Rokhlin property in terms of asymptotic retracts. As a consequence, we derive strong $KK$-theoretical obstructions to the continuous Rokhlin property. Using these, we show that the Universal Coefficient Theorem (UCT) is preserved under formation of crossed products and passage to fixed point algebras by such actions, even in the absence of nuclearity. As an application of the case of ${{\mathbb{Z}}}_3$-actions, we answer a question of Phillips–Viola about algebras not isomorphic to their opposites. Our analysis of the $KK$-theory of the crossed product allows us to prove a ${{\mathbb{T}}}$-equivariant version of Kirchberg–Phillips: two circle actions with the continuous Rokhlin property on Kirchberg algebras are conjugate whenever they are $KK^{{{\mathbb{T}}}}$-equivalent. In the presence of the UCT, this is equivalent to having isomorphic equivariant $K$-theory. We moreover characterize the $KK^{{{\mathbb{T}}}}$-theoretical invariants that arise in this way. Finally, we identify a $KK^{{{\mathbb{T}}}}$-theoretic obstruction to the continuous property, which is shown to be the only obstruction in the setting of Kirchberg algebras. We show by means of explicit examples that the Rokhlin property is strictly weaker than the continuous Rokhlin property.

scholarly journals Fullness of crossed products of factors by discrete groups

2019 ◽  
Vol 150 (5) ◽  
pp. 2368-2378 ◽  
Author(s):  
Amine Marrakchi
Keyword(s):  
Discrete Group ◽  
Amenable Group ◽  
Complete Characterization ◽  
Crossed Product ◽  
Discrete Groups ◽  
Sufficient Condition ◽  
Crossed Products ◽  
Product Factor ◽  
Jones Criterion ◽  
Arbitrary Factor

AbstractLet M be an arbitrary factor and $\sigma : \Gamma \curvearrowright M$ an action of a discrete group. In this paper, we study the fullness of the crossed product $M \rtimes _\sigma \Gamma $. When Γ is amenable, we obtain a complete characterization: the crossed product factor $M \rtimes _\sigma \Gamma $ is full if and only if M is full and the quotient map $\overline {\sigma } : \Gamma \rightarrow {\rm out}(M)$ has finite kernel and discrete image. This answers the question of Jones from [11]. When M is full and Γ is arbitrary, we give a sufficient condition for $M \rtimes _\sigma \Gamma $ to be full which generalizes both Jones' criterion and Choda's criterion. In particular, we show that if M is any full factor (possibly of type III) and Γ is a non-inner amenable group, then the crossed product $M \rtimes _\sigma \Gamma $ is full.

scholarly journals Exel's crossed product for non-unital C*-algebras

2010 ◽  
Vol 149 (3) ◽  
pp. 423-444 ◽  
Author(s):  
NATHAN BROWNLOWE ◽  
IAIN RAEBURN ◽  
SEAN T. VITTADELLO
Keyword(s):  
Dynamical Systems ◽  
Transfer Operator ◽  
Finite Graph ◽  
Crossed Product ◽  
Crossed Products ◽  
Ideal Structure ◽  
Locally Finite ◽  
C Algebras ◽  
Locally Finite Graph ◽  
The Ideal

AbstractWe consider a family of dynamical systems (A, α, L) in which α is an endomorphism of a C*-algebra A and L is a transfer operator for α. We extend Exel's construction of a crossed product to cover non-unital algebras A, and show that the C*-algebra of a locally finite graph can be realised as one of these crossed products. When A is commutative, we find criteria for the simplicity of the crossed product, and analyse the ideal structure of the crossed product.

scholarly journals On maximal ideals in certain reduced twisted C*-crossed products

2015 ◽  
Vol 158 (3) ◽  
pp. 399-417 ◽  
Author(s):  
ERIK BÉDOS ◽  
ROBERTO CONTI
Keyword(s):  
Discrete Group ◽  
Crossed Product ◽  
Crossed Products ◽  
Bijective Correspondence ◽  
C Algebra ◽  
Maximal Ideals ◽  
Maximal Invariant

AbstractWe consider a twisted action of a discrete groupGon a unital C*-algebraAand give conditions ensuring that there is a bijective correspondence between the maximal invariant ideals ofAand the maximal ideals in the associated reduced C*-crossed product.

scholarly journals Contractive Spectral Triples for Crossed Products

2014 ◽  
Vol 114 (2) ◽  
pp. 275 ◽  
Author(s):  
Alan L. T. Paterson
Keyword(s):  
Coding Theory ◽  
Duality Theorem ◽  
Discrete Group ◽  
Crossed Product ◽  
Contractive Condition ◽  
Crossed Products ◽  
Isometric Condition ◽  
Spectral Triples ◽  
Groups Of Diffeomorphisms ◽  
Discrete Group Action

Connes showed that spectral triples encode (noncommutative) metric information. Further, Connes and Moscovici in their metric bundle construction showed that, as with the Takesaki duality theorem, forming a crossed product spectral triple can substantially simplify the structure. In a recent paper, Bellissard, Marcolli and Reihani (among other things) studied in depth metric notions for spectral triples and crossed product spectral triples for $Z$-actions, with applications in number theory and coding theory. In the work of Connes and Moscovici, crossed products involving groups of diffeomorphisms and even of étale groupoids are required. With this motivation, the present paper develops part of the Bellissard-Marcolli-Reihani theory for a general discrete group action, and in particular, introduces coaction spectral triples and their associated metric notions. The isometric condition is replaced by the contractive condition.

scholarly journals Strong pure infiniteness of crossed products

2016 ◽  
Vol 38 (1) ◽  
pp. 220-243
Author(s):  
E. KIRCHBERG ◽  
A. SIERAKOWSKI
Keyword(s):  
Discrete Group ◽  
Closed Ideal ◽  
Crossed Product ◽  
General Criterion ◽  
Crossed Products ◽  
Compact Sets ◽  
Compact Spaces ◽  
Is Element ◽  
Contraction Property ◽  
Disjoint Sets

Consider an exact action of a discrete group $G$ on a separable C*-algebra $A$. It is shown that the reduced crossed product $A\rtimes _{\unicode[STIX]{x1D70E},\unicode[STIX]{x1D706}}G$ is strongly purely infinite—provided that the action of $G$ on any quotient $A/I$ by a $G$-invariant closed ideal $I\neq A$ is element-wise properly outer and that the action of $G$ on $A$ is $G$-separating (cf. Definition 5.1). This is the first non-trivial sufficient general criterion for strong pure infiniteness of reduced crossed products of C*-algebras $A$ that are not $G$-simple. In the case $A=\text{C}_{0}(X)$, the notion of a $G$-separating action corresponds to the property that two compact sets $C_{1}$ and $C_{2}$, that are contained in open subsets $C_{j}\subseteq U_{j}\subseteq X$, can be mapped by elements $g_{1},g_{2}$ of $G$ onto disjoint sets $\unicode[STIX]{x1D70E}_{g_{j}}(C_{j})\subseteq U_{j}$, but satisfy not necessarily the contraction property $\unicode[STIX]{x1D70E}_{g_{j}}(U_{j})\subseteq \overline{U_{j}}$. A generalization of strong boundary actions on compact spaces to non-unital and non-commutative C*-algebras $A$ (cf. Definition 7.1) is also introduced. It is stronger than the notion of $G$-separating actions by Proposition 7.6, because $G$-separation does not imply $G$-simplicity and there are examples of $G$-separating actions with reduced crossed products that are stably projection-less and non-simple.

A Galois Correspondence for Reduced Crossed Products of Simple -algebras by Discrete Groups

2019 ◽  
Vol 71 (5) ◽  
pp. 1103-1125 ◽  
Author(s):  
Jan Cameron ◽  
Roger R. Smith
Keyword(s):  
Dynamical System ◽  
Discrete Group ◽  
Crossed Product ◽  
Discrete Groups ◽  
Crossed Products ◽  
Outer Automorphisms ◽  
Galois Correspondence ◽  
Simple Algebras ◽  
Unital Simple

AbstractLet a discrete group $G$ act on a unital simple $\text{C}^{\ast }$-algebra $A$ by outer automorphisms. We establish a Galois correspondence $H\mapsto A\rtimes _{\unicode[STIX]{x1D6FC},r}H$ between subgroups of $G$ and $\text{C}^{\ast }$-algebras $B$ satisfying $A\subseteq B\subseteq A\rtimes _{\unicode[STIX]{x1D6FC},r}G$, where $A\rtimes _{\unicode[STIX]{x1D6FC},r}G$ denotes the reduced crossed product. For a twisted dynamical system $(A,G,\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D70E})$, we also prove the corresponding result for the reduced twisted crossed product $A\rtimes _{\unicode[STIX]{x1D6FC},r}^{\unicode[STIX]{x1D70E}}G$.

scholarly journals Strong Morita equivalence for inclusions of $C^*$-algebras induced by twisted actions of a countable discrete group

2021 ◽  
Vol 127 (2) ◽  
pp. 317-336
Author(s):  
Kazunori Kodaka
Keyword(s):  
Discrete Group ◽  
Morita Equivalence ◽  
Crossed Product ◽  
Crossed Products ◽  
C Algebra ◽  
C Algebras ◽  
Morita Equivalent ◽  
Countable Discrete Group ◽  
Relative Commutant ◽  
Strong Morita Equivalence

We consider two twisted actions of a countable discrete group on $\sigma$-unital $C^*$-algebras. Then by taking the reduced crossed products, we get two inclusions of $C^*$-algebras. We suppose that they are strongly Morita equivalent as inclusions of $C^*$-algebras. Also, we suppose that one of the inclusions of $C^*$-algebras is irreducible, that is, the relative commutant of one of the $\sigma$-unital $C^*$-algebra in the multiplier $C^*$-algebra of the reduced twisted crossed product is trivial. We show that the two actions are then strongly Morita equivalent up to some automorphism of the group.

Decomposition theorems for certain C*-crossed products

1983 ◽  
Vol 94 (2) ◽  
pp. 265-275 ◽  
Author(s):  
Marc de Brabanter
Keyword(s):  
Dynamical System ◽  
Discrete Group ◽  
Crossed Product ◽  
Crossed Products ◽  
Decomposition Theorems

Let G be an abelian discrete group, A a unital C*-algebra and an action of G on A, i.e. (A, G,) is a C*-dynamical system. Let K denote the kernel ker of and put R = G/K. The main purpose of this article is to determine the roles of K and R in the crossed product G A. This goal is achieved in Section 2, where we prove that G A is *-isomorphic to a twisted crossed product of R with C*(K) A with respect to the action 1 and a 2-cocycle related to the 2-cocycle determined by the extension G of R by K. Here is the obvious action of R on A.

Sign in / Sign up

Export Citation Format

Share Document