A characterization of semi-crossed products of finite-dimensional C*-algebras

AbstractLet A and B be arbitrary C*-algebras, we prove that the existence of a Hilbert A–B-bimodule of finite index ensures that the WEP, QWEP, and LLP along with other finite-dimensional approximation properties such as CBAP and (S)OAP are shared by A and B. For this, we first study the stability of the WEP, QWEP, and LLP under Morita equivalence of C*-algebras. We present examples of Hilbert A–B-bimodules, which are not of finite index, while such properties are shared between A and B. To this end, we study twisted crossed products by amenable discrete groups.

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A Continuous Field of Projectionless C*-Algebras

Canadian Journal of Mathematics ◽

10.4153/cjm-2001-003-2 ◽

2001 ◽

Vol 53 (1) ◽

pp. 51-72 ◽

Cited By ~ 4

Author(s):

Andrew Dean

Keyword(s):

Continuous Functions ◽

Unit Interval ◽

Type I ◽

Crossed Products ◽

Inductive Limits ◽

Direct Sums ◽

Matrix Algebras ◽

C Algebras ◽

Finite Dimensional ◽

Continuous Field

AbstractWe use some results about stable relations to show that some of the simple, stable, projectionless crossed products of O2 by considered by Kishimoto and Kumjian are inductive limits of type C*-algebras. The type I C*-algebras that arise are pullbacks of finite direct sums of matrix algebras over the continuous functions on the unit interval by finite dimensional C*-algebras.

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Equivariant KK-Theory and the Continuous Rokhlin Property

International Mathematics Research Notices ◽

10.1093/imrn/rnaa292 ◽

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.

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MINIMAL DYNAMICS AND $\mathcal{Z}$-STABLE CLASSIFICATION

International Journal of Mathematics ◽

10.1142/s0129167x10006665 ◽

2011 ◽

Vol 22 (01) ◽

pp. 1-23 ◽

Cited By ~ 8

Author(s):

KAREN R. STRUNG ◽

WILHELM WINTER

Keyword(s):

Dynamical Systems ◽

Compact Metric Space ◽

C Algebra ◽

C Algebras ◽

Finite Dimensional ◽

Minimal Dynamical Systems ◽

Uniquely Ergodic ◽

Stable Classification

Let X be an infinite compact metric space, α : X → X a minimal homeomorphism, u the unitary that implements α in the transformation group C*-algebra C(X) ⋊α ℤ, and [Formula: see text] a class of separable nuclear C*-algebras that contains all unital hereditary C*-subalgebras of C*-algebras in [Formula: see text]. Motivated by the success of tracial approximation by finite dimensional C*-algebras as an abstract characterization of classifiable C*-algebras and the idea that classification results for C*-algebras tensored with UHF algebras can be used to derive classification results up to tensoring with the Jiang-Su algebra [Formula: see text], we prove that (C(X) ⋊α ℤ) ⊗ Mq∞ is tracially approximately [Formula: see text] if there exists a y ∈ X such that the C*-subalgebra (C*(C(X), uC0(X\{y}))) ⊗ Mq∞ is tracially approximately [Formula: see text]. If the class [Formula: see text] consists of finite dimensional C*-algebras, this can be used to deduce classification up to tensoring with [Formula: see text] for C*-algebras associated to minimal dynamical systems where projections separate tracial states. This is done without making any assumptions on the real rank or stable rank of either C(X) ⋊α ℤ or C*(C(X), uC0(X\{y})), nor on the dimension of X. The result is a key step in the classification of C*-algebras associated to uniquely ergodic minimal dynamical systems by their ordered K-groups. It also sets the stage to provide further classification results for those C*-algebras of minimal dynamical systems where projections do not necessarily separate traces.

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Full duality for coactions of discrete groups

MATHEMATICA SCANDINAVICA ◽

10.7146/math.scand.a-14374 ◽

2002 ◽

Vol 90 (2) ◽

pp. 267 ◽

Cited By ~ 7

Author(s):

Siegfried Echterhoff ◽

John Quigg

Keyword(s):

Discrete Group ◽

Discrete Groups ◽

Crossed Products ◽

Normal Subgroups ◽

C Algebras ◽

Full Duality ◽

Dual Action ◽

Strong Relation ◽

Usual Theory

Using the strong relation between coactions of a discrete group $G$ on $C^*$-algebras and Fell bundles over $G$ we prove a new version of Mansfield's imprimitivity theorem for coactions of discrete groups. Our imprimitivity theorem works for the universally defined full crossed products and arbitrary subgroups of $G$ as opposed to the usual theory of [16], [11] which uses the spatially defined reduced crossed products and normal subgroups of $G$. Moreover, our theorem factors through the usual one by passing to appropriate quotients. As applications we show that a Fell bundle over a discrete group is amenable in the sense of Exel [7] if and only if the double dual action is amenable in the sense that the maximal and reduced crossed products coincide. We also give a new characterization of induced coactions in terms of their dual actions.

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Classification of Integrodifferential C-Algebras*

Symmetry ◽

10.3390/sym13101900 ◽

2021 ◽

Vol 13 (10) ◽

pp. 1900

Author(s):

Anton A. Kutsenko

Keyword(s):

Infinite Product ◽

Complete Characterization ◽

Difference Operators ◽

C Algebra ◽

C Algebras ◽

Finite Dimensional ◽

Differential Algebras ◽

Af Algebras ◽

Product Of Matrices

The infinite product of matrices with integer entries, known as a modified Glimm–Bratteli symbol n, is a new, sufficiently simple, and very powerful tool for the characterization of approximately finite-dimensional (AF) algebras. This symbol provides a convenient algebraic representation of the Bratteli diagram for AF algebras in the same way as was previously performed by J. Glimm for more simple uniformly hyperfinite (UHF) algebras. We apply this symbol to characterize integrodifferential algebras. The integrodifferential algebra FN,M is the C*-algebra generated by the following operators acting on L2([0,1)N→CM): (1) operators of multiplication by bounded matrix-valued functions, (2) finite-difference operators, and (3) integral operators. Most of the operators and their approximations studying in physics belong to these algebras. We give a complete characterization of FN,M. In particular, we show that FN,M does not depend on M, but depends on N. At the same time, it is known that differential algebras HN,M, generated by the operators (1) and (2) only, do not depend on both dimensions N and M; they are all *-isomorphic to the universal UHF algebra. We explicitly compute the Glimm–Bratteli symbols (for HN,M, it was already computed earlier) which completely characterize the corresponding AF algebras. This symbol n is an infinite product of matrices with nonnegative integer entries. Roughly speaking, all the symmetries appearing in the approximation of complex infinite-dimensional integrodifferential and differential algebras by finite-dimensional ones are coded by a product of integer matrices.

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Crossed products of C∗-algebras C(X,A) and their applications

International Journal of Mathematics ◽

10.1142/s0129167x16500294 ◽

2016 ◽

Vol 27 (03) ◽

pp. 1650029

Author(s):

Jiajie Hua

Keyword(s):

Simple Formula ◽

Continuous Functions ◽

Crossed Products ◽

Compact Metric Space ◽

Covering Dimension ◽

C Algebras ◽

Finite Dimensional ◽

Tracial Rank ◽

Stably Finite ◽

Unital Simple

Let [Formula: see text] be an infinite compact metric space with finite covering dimension, let [Formula: see text] be a unital separable simple AH-algebra with no dimension growth, and denote by [Formula: see text] the [Formula: see text]-algebra of all continuous functions from [Formula: see text] to [Formula: see text] Suppose that [Formula: see text] is a minimal group action and the induced [Formula: see text]-action on [Formula: see text] is free. Under certain conditions, we show the crossed product [Formula: see text]-algebra [Formula: see text] has rational tracial rank zero and hence is classified by its Elliott invariant. Next, we show the following: Let [Formula: see text] be a Cantor set, let [Formula: see text] be a stably finite unital separable simple [Formula: see text]-algebra which is rationally TA[Formula: see text] where [Formula: see text] is a class of separable unital [Formula: see text]-algebras which is closed under tensoring with finite dimensional [Formula: see text]-algebras and closed under taking unital hereditary sub-[Formula: see text]-algebras, and let [Formula: see text]. Under certain conditions, we conclude that [Formula: see text] is rationally TA[Formula: see text] Finally, we classify the crossed products of certain unital simple [Formula: see text]-algebras by using the crossed products of [Formula: see text].

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FINITE-DIMENSIONAL REPRESENTATIONS OF FREE PRODUCT C*-ALGEBRAS

International Journal of Mathematics ◽

10.1142/s0129167x92000217 ◽

1992 ◽

Vol 03 (04) ◽

pp. 469-476 ◽

Cited By ~ 18

Author(s):

RUY EXEL ◽

TERRY A. LORING

Keyword(s):

Free Product ◽

C Algebras ◽

Finite Dimensional

Our main theorem is a characterization of C*-algebras that have a separating family of finite-dimensional representations. This characterization makes possible a solution to a problem posed by Goodearl and Menaul. Specifically, we prove that the free product of such C*-algebras again has this property.

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Boundary Action On Simple Reduced Group C*-Algebras

The American Journal of Interdisciplinary Innovations and Research ◽

10.37547/tajiir/volume03issue01-01 ◽

2021 ◽

Vol 03 (01) ◽

pp. 1-15

Author(s):

V.A. Onuche ◽

◽

T.J Alabiand ◽

O.F. Ajayi ◽

◽

...

Keyword(s):

Simple Group ◽

Free Group ◽

Discrete Group ◽

Crossed Products ◽

Ideal Structure ◽

Stability Properties ◽

C Algebras ◽

Reduced Group ◽

The Stability

A connection between boundary actions, ideal structure of reduced crossed products and C*-simple group is imminent.We investigate the stability properties for discrete group pioneered by powers and show that the non-abelian free group on two generators is C*-simple.Kalantar and Kennedy [32, Theorem 6.2] is now extended. Some examples are given using characterization of C*-simplicity obtained by Kalantar, Kennedy, Breuillard, and Ozawa [10, Theorem 3.1]

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