scholarly journals Residually finite actions and crossed products

2011 ◽  
Vol 32 (5) ◽  
pp. 1585-1614 ◽  
Author(s):  
DAVID KERR ◽  
PIOTR W. NOWAK
Keyword(s):  
Free Group ◽  
Metrizable Space ◽  
Crossed Product ◽  
Discrete Groups ◽  
Crossed Products ◽  
Residual Finiteness ◽  
Hausdorff Spaces ◽  
Residually Finite ◽  
Compact Metrizable Space ◽  
Continuous Actions

AbstractWe study a notion of residual finiteness for continuous actions of discrete groups on compact Hausdorff spaces and how it relates to the existence of norm microstates for the reduced crossed product. Our main result asserts that an action of a free group on a zero-dimensional compact metrizable space is residually finite if and only if its reduced crossed product admits norm microstates, i.e., is an MF algebra.

scholarly journals Crossed products of totally disconnected spaces by

1993 ◽  
Vol 13 (3) ◽  
pp. 445-484 ◽  
Author(s):  
Ola Bratteli ◽  
David E. Evans ◽  
Akitaka Kishimoto
Keyword(s):  
Fixed Point ◽  
Metrizable Space ◽  
Crossed Product ◽  
Crossed Products ◽  
Prove Theorem ◽  
Compact Metrizable Space ◽  
Totally Disconnected

AbstractLet Ω be a totally disconnected compact metrizable space, and let α be a minimal homeomorphism of Ω. Let σ be a homeomorphism of order 2 on Ω such that ασ = σα−1, and assume that σ or ασ has a fixed point. We prove (Theorem 3.5) that the crossed product is an AF-algebra.

scholarly journals Supramenable groups and partial actions

2016 ◽  
Vol 37 (5) ◽  
pp. 1592-1606 ◽  
Author(s):  
EDUARDO P. SCARPARO
Keyword(s):  
Semidirect Product ◽  
Crossed Product ◽  
Probability Measures ◽  
Crossed Products ◽  
Hausdorff Spaces ◽  
Tracial States ◽  
Product Of Groups

We characterize supramenable groups in terms of the existence of invariant probability measures for partial actions on compact Hausdorff spaces and the existence of tracial states on partial crossed products. These characterizations show that, in general, one cannot decompose a partial crossed product of a $\text{C}^{\ast }$-algebra by a semidirect product of groups into two iterated partial crossed products. However, we give conditions which ensure that such decomposition is possible.

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 Isometries of Spaces of Radon Measures

2017 ◽  
Vol 2017 ◽  
pp. 1-4
Author(s):  
Marek Wójtowicz
Keyword(s):  
Closed Subset ◽  
Unit Interval ◽  
Metrizable Space ◽  
Discrete Space ◽  
Hausdorff Spaces ◽  
Bernstein Type ◽  
Radon Measures ◽  
Compact Metrizable Space

Let Ω and I denote a compact metrizable space with card(Ω)≥2 and the unit interval, respectively. We prove Milutin and Cantor-Bernstein type theorems for the spaces M(Ω) of Radon measures on compact Hausdorff spaces Ω. In particular, we obtain the following results: (1) for every infinite closed subset K of βN the spaces M(K), M(βN), and M(Ω2ℵ0) are order-isometric; (2) for every discrete space Γ with m≔card(Γ)>ℵ0 the spaces M(βΓ) and M(I2m) are order-isometric, whereas there is no linear homeomorphic injection from C(βT) into C(I2m).

scholarly journals TENSOR-PRODUCT COACTION FUNCTORS

2020 ◽  
pp. 1-16
Author(s):  
S. KALISZEWSKI ◽  
MAGNUS B. LANDSTAD ◽  
JOHN QUIGG
Keyword(s):  
Tensor Product ◽  
Recent Work ◽  
Analogous Result ◽  
Crossed Product ◽  
Discrete Groups ◽  
Crossed Products ◽  
Fixed Action ◽  
Product Action ◽  
K Theory

Recent work by Baum et al. [‘Expanders, exact crossed products, and the Baum–Connes conjecture’, Ann. K-Theory 1(2) (2016), 155–208], further developed by Buss et al. [‘Exotic crossed products and the Baum–Connes conjecture’, J. reine angew. Math. 740 (2018), 111–159], introduced a crossed-product functor that involves tensoring an action with a fixed action $(C,\unicode[STIX]{x1D6FE})$ , then forming the image inside the crossed product of the maximal-tensor-product action. For discrete groups, we give an analogue for coaction functors. We prove that composing our tensor-product coaction functor with the full crossed product of an action reproduces their tensor-crossed-product functor. We prove that every such tensor-product coaction functor is exact, and if $(C,\unicode[STIX]{x1D6FE})$ is the action by translation on $\ell ^{\infty }(G)$ , we prove that the associated tensor-product coaction functor is minimal, thereby recovering the analogous result by the above authors. Finally, we discuss the connection with the $E$ -ization functor we defined earlier, where $E$ is a large ideal of $B(G)$ .

scholarly journals On the topological stable rank of certain transformation group C*-algebras

1990 ◽  
Vol 10 (1) ◽  
pp. 197-207 ◽  
Author(s):  
Ian F. Putnam
Keyword(s):  
Transformation Group ◽  
Metrizable Space ◽  
Stable Rank ◽  
Residual Finiteness ◽  
Dynamical Condition ◽  
C Algebras ◽  
Compact Metrizable Space ◽  
Topological Stable Rank ◽  
Necessary And Sufficient ◽  
Totally Disconnected

AbstractWe consider the crossed product or transformation group C*-algebras arising from actions of the group of integers on a totally disconnected compact metrizable space. Under a mild hypothesis, we give a necessary and sufficient dynamical condition for the invertibles in such a C*-algebra to be dense. We also examine the property of residual finiteness for such C*-algebras.

scholarly journals Residual finiteness and ‘free’ distributively generated near-rings

1979 ◽  
Vol 28 (4) ◽  
pp. 398-400 ◽  
Author(s):  
David J. John
Keyword(s):  
Free Group ◽  
Finite Semigroup ◽  
Residual Finiteness ◽  
Residually Finite ◽  
Variety Of Groups

AbstractLet V be a variety of groups in which the free group is residually finite, and let S be a residually finite semigroup. Let Nv(S) be the ‘free’ distributively generated near-ring constructed from S and V. Theorem; Nv(S) is residually finite.

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 C*-crossed products by partial actions and actions of inverse semigroups

1997 ◽  
Vol 63 (1) ◽  
pp. 32-46 ◽  
Author(s):  
Nándor Sieben
Keyword(s):  
Inverse Semigroups ◽  
Crossed Product ◽  
Discrete Groups ◽  
Crossed Products ◽  
Related Concept ◽  
Semigroup Action ◽  
Covariant Representations

AbstractThe recently developed theory of partial actions of discrete groups on C*-algebras is extended. A related concept of actions of inverse semigroups on C*-algebras is defined, including covariant representations and crossed products. The main result is that every partial crossed product is a crossed product by a semigroup action.

Embedding covariance algebras of flows into $AF$-algebras

1999 ◽  
Vol 19 (3) ◽  
pp. 723-740 ◽  
Author(s):  
MICHAEL V. PIMSNER
Keyword(s):  
Metrizable Space ◽  
Crossed Product ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Injective Homomorphism ◽  
Compact Metrizable Space ◽  
Necessary And Sufficient ◽  
Af Algebras

Suppose that $\{\alpha_t\}_{t\in \mathbb{R}}$ is a flow on the compact metrizable space $X$. We prove that a necessary and sufficient condition for the existence of an embedding (injective $*$-homomorphism) of the crossed product $C(X)\rtimes_\alpha \mathbb{R}$ into some $AF$-algebra is that every point of $X$ be chain recurrent in the sense of Conley.

Sign in / Sign up

Export Citation Format

Share Document