scholarly journals Primeness and Primitivity Conditions for Twisted Group $C^*$-Algebras

2014 ◽  
Vol 114 (2) ◽  
pp. 299 ◽  
Author(s):  
Tron Ånen Omland
Keyword(s):  
Discrete Group ◽  
Direct Products ◽  
Free Products ◽  
C Algebra ◽  
C Algebras ◽  
Twisted Group ◽  
Products Of Groups

For a multiplier (2-cocycle) $\sigma$ on a discrete group $G$ we give conditions for which the twisted group $C^*$-algebra associated with the pair $(G,\sigma)$ is prime or primitive. We also discuss how these conditions behave on direct products and free products of groups.

Property T and Amenable Transformation Group C*-algebras

2015 ◽  
Vol 58 (1) ◽  
pp. 110-114 ◽  
Author(s):  
F. Kamalov
Keyword(s):  
Compact Hausdorff Space ◽  
Discrete Group ◽  
Hausdorff Space ◽  
Transformation Group ◽  
Amenable Group ◽  
Transformation Groups ◽  
C Algebra ◽  
C Algebras ◽  
Property T ◽  
Tracial States

AbstractIt is well known that a discrete group that is both amenable and has Kazhdan’s Property T must be finite. In this note we generalize this statement to the case of transformation groups. We show that if G is a discrete amenable group acting on a compact Hausdorff space X, then the transformation group C*-algebra C*(X; G) has Property T if and only if both X and G are finite. Our approach does not rely on the use of tracial states on C*(X; G).

scholarly journals Conditions for primitivity of unital amalgamated full free products of finite-dimensional C*-algebras

2014 ◽  
Vol 25 (09) ◽  
pp. 1450086
Author(s):  
Francisco Torres-Ayala
Keyword(s):  
Free Products ◽  
C Algebra ◽  
C Algebras ◽  
Finite Dimensional ◽  
Trivial Center

We consider amalgamated unital full free products of the form A1*D A2, where A1, A2 and D are finite-dimensional C*-algebras and there are faithful traces on A1 and A2 whose restrictions to D agree. We provide several conditions on the matrices of partial multiplicities of the inclusions D ↪ A1 and D ↪ A2 that guarantee that the C*-algebra A1*D A2 is primitive. If the ranks of the matrices of partial multiplicities are one or all entries are 0 or ≥ 2, we prove that the algebra A1*D A2 is primitive if and only if it has a trivial center.

scholarly journals A Lifting Characterization of Rfd C*-Algebras

2014 ◽  
Vol 115 (1) ◽  
pp. 85 ◽  
Author(s):  
Don Hadwin
Keyword(s):  
Hilbert Spaces ◽  
Infinite Cardinal ◽  
Free Products ◽  
C Algebra ◽  
C Algebras ◽  
The Family

We prove a conjecture of Terry Loring that characterizes separable RFD C*-algebras in terms of a lifting property. In addition we introduce and study generalizations of RFD algebras. If $k$ is an infinite cardinal, we say a C*-algebra is residually less than $k$ dimensional, if the family of representations on Hilbert spaces of dimension less than $k$ separates the points of the algebra. We give characterizations of this property and prove that this class is closed under free products in the nonunital category. For free products in the unital category, the results depend on the cardinal $k$.

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.

scholarly journals C∗-algebras associated with free products of groups

1979 ◽  
Vol 82 (1) ◽  
pp. 211-221 ◽  
Author(s):  
William Paschke ◽  
Norberto Salinas
Keyword(s):  
Free Products ◽  
C Algebras ◽  
Products Of Groups

scholarly journals Topological Free Entropy Dimensions in Nuclear C*-algebras and in Full Free Products of Unital C*-algebras

2011 ◽  
Vol 63 (3) ◽  
pp. 551-590 ◽  
Author(s):  
Don Hadwin ◽  
Qihui Li ◽  
Junhao Shen
Keyword(s):  
Free Product ◽  
Finite Family ◽  
Free Products ◽  
C Algebra ◽  
Free Entropy ◽  
C Algebras ◽  
Entropy Dimension ◽  
Free Entropy Dimension

Abstract In the paper, we introduce a new concept, topological orbit dimension of an n-tuple of elements in a unital C*-algebra. Using this concept, we conclude that Voiculescu's topological free entropy dimension of every finite family of self-adjoint generators of a nuclear C*-algebra is less than or equal to 1. We also show that the Voiculescu's topological free entropy dimension is additive in the full free product of some unital C*-algebras. We show that the unital full free product of Blackadar and Kirchberg's unital MF algebras is also an MF algebra. As an application, we obtain that Ext(C*r (F2) *C C* r (F2)) is not a group.

scholarly journals EXACTNESS OF CUNTZ–PIMSNER C*-ALGEBRAS

2001 ◽  
Vol 44 (2) ◽  
pp. 425-444 ◽  
Author(s):  
Kenneth J. Dykema ◽  
Dimitri Shlyakhtenko
Keyword(s):  
Topological Entropy ◽  
Subject Classification ◽  
Primary 46L08 ◽  
Free Products ◽  
C Algebra ◽  
C Algebras ◽  
Amalgamated Free Products ◽  
Finite Dimensional

AbstractLet $H$ be a full Hilbert bimodule over a $C^*$-algebra $A$. We show that the Cuntz–Pimsner algebra associated to $H$ is exact if and only if $A$ is exact. Using this result, we give alternative proofs for exactness of reduced amalgamated free products of exact $C^*$-algebras. In the case in which $A$ is a finite-dimensional $C^*$-algebra, we also show that the Brown–Voiculescu topological entropy of Bogljubov automorphisms of the Cuntz–Pimsner algebra associated to an $A,A$ Hilbert bimodule is zero.AMS 2000 Mathematics subject classification: Primary 46L08. Secondary 46L09; 46L54

scholarly journals Nuclear subalgebras of UHF C*-algebras

1986 ◽  
Vol 29 (1) ◽  
pp. 97-100 ◽  
Author(s):  
R. J. Archbold ◽  
Alexander Kumjian
Keyword(s):  
Tensor Product ◽  
Inductive Limit ◽  
Inductive Limits ◽  
C Algebra ◽  
C Algebras ◽  
Finite Dimensional ◽  
The Family ◽  
Algebraic Tensor Product

A C*-algebra A is said to be approximately finite dimensional (AF) if it is the inductive limit of a sequence of finite dimensional C*-algebras(see [2], [5]). It is said to be nuclear if, for each C*-algebra B, there is a unique C*-norm on the *-algebraic tensor product A ⊗B [11]. Since finite dimensional C*-algebras are nuclear, and inductive limits of nuclear C*-algebras are nuclear [16];,every AF C*-algebra is nuclear. The family of nuclear C*-algebras is a large and well-behaved class (see [12]). The AF C*-algebras for a particularly tractable sub-class which has been completely classified in terms of the invariant K0 [7], [5].

scholarly journals Free product decompositions in images of certain free products of groups

2007 ◽  
Vol 310 (1) ◽  
pp. 57-69
Author(s):  
N.S. Romanovskii ◽  
John S. Wilson
Keyword(s):  
Free Product ◽  
Free Products ◽  
Products Of Groups

On C*-Algebras Associated with Subshifts

1997 ◽  
Vol 08 (03) ◽  
pp. 357-374 ◽  
Author(s):  
Kengo Matsumoto
Keyword(s):  
Symbolic Dynamics ◽  
Markov Shift ◽  
C Algebra ◽  
C Algebras ◽  
Markov Shifts ◽  
Universal Properties

We construct and study C*-algebras associated with subshifts in symbolic dynamics as a generalization of Cuntz–Krieger algebras for topological Markov shifts. We prove some universal properties for the C*-algebras and give a criterion for them to be simple and purely infinite. We also present an example of a C*-algebra coming from a subshift which is not conjugate to a Markov shift.

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