Group actions on Smale space -algebras

2019 ◽  
Vol 40 (9) ◽  
pp. 2368-2398
Author(s):  
ROBIN J. DEELEY ◽  
KAREN R. STRUNG
Keyword(s):  
Dynamical System ◽  
Effective Action ◽  
Discrete Group ◽  
Group Actions ◽  
Crossed Products ◽  
Outer Action ◽  
Smale Space ◽  
Smale Spaces

Group actions on a Smale space and the actions induced on the $\text{C}^{\ast }$-algebras associated to such a dynamical system are studied. We show that an effective action of a discrete group on a mixing Smale space produces a strongly outer action on the homoclinic algebra. We then show that for irreducible Smale spaces, the property of finite Rokhlin dimension passes from the induced action on the homoclinic algebra to the induced actions on the stable and unstable $\text{C}^{\ast }$-algebras. In each of these cases, we discuss the preservation of properties (such as finite nuclear dimension, ${\mathcal{Z}}$-stability, and classification by Elliott invariants) in the resulting crossed products.

scholarly journals Smale spaces via inverse limits

2013 ◽  
Vol 34 (6) ◽  
pp. 2066-2092 ◽  
Author(s):  
SUSANA WIELER
Keyword(s):  
Dynamical System ◽  
Inverse Limits ◽  
Canonical Coordinates ◽  
Chaotic Dynamical System ◽  
Smale Space ◽  
Smale Spaces ◽  
Special Case ◽  
Basic Sets

AbstractA Smale space is a chaotic dynamical system with canonical coordinates of contracting and expanding directions. The basic sets for Smale’s Axiom $A$ systems are a key class of examples. We consider the special case of irreducible Smale spaces with zero-dimensional contracting directions, and characterize these as stationary inverse limits satisfying certain conditions.

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 $A\rtimes _{\unicode[STIX]{x1D6FC},r}^{\unicode[STIX]{x1D70E}}G$.

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.

Markov partitions and homology for -solenoids

2015 ◽  
Vol 37 (3) ◽  
pp. 716-738 ◽  
Author(s):  
NIGEL D. BURKE ◽  
IAN F. PUTNAM
Keyword(s):  
Dynamical System ◽  
Homology Theory ◽  
Prime Pair ◽  
Topological Dynamical System ◽  
Markov Partitions ◽  
Local Coordinates ◽  
Smale Space ◽  
The Given ◽  
Smale Spaces ◽  
Special Case

Given a relatively prime pair of integers, $n\geq m>1$, there is associated a topological dynamical system which we refer to as an $n/m$-solenoid. It is also a Smale space, as defined by David Ruelle, meaning that it has local coordinates of contracting and expanding directions. In this case, these are locally products of the real and various $p$-adic numbers. In the special case, $m=2,n=3$ and for $n>3m$, we construct Markov partitions for such systems. The second author has developed a homology theory for Smale spaces and we compute this in these examples, using the given Markov partitions, for all values of $n\geq m>1$ and relatively prime.

The Baum–Connes Conjecture and Discrete Group Actions on Trees

K-Theory ◽  
1999 ◽  
Vol 17 (4) ◽  
pp. 303-318 ◽  
Author(s):  
Jean-louis Tu
Keyword(s):  
Discrete Group ◽  
Group Actions

scholarly journals Discrete product systems with twisted units

1995 ◽  
Vol 52 (2) ◽  
pp. 317-326 ◽  
Author(s):  
Marcelo Laca
Keyword(s):  
Dynamical System ◽  
Crossed Product ◽  
Type I ◽  
Crossed Products ◽  
Product System ◽  
C Algebra ◽  
Covariant Representations ◽  
Product Systems ◽  
Recent Theorem ◽  
I Factor

The spectral C*-algebra of the discrete product systems of H.T. Dinh is shown to be a twisted semigroup crossed product whenever the product system has a twisted unit. The covariant representations of the corresponding dynamical system are always faithful, implying the simplicity of these crossed products; an application of a recent theorem of G.J. Murphy gives their nuclearity. Furthermore, a semigroup of endomorphisms of B(H) having an intertwining projective semigroup of isometries can be extended to a group of automorphisms of a larger Type I factor.

scholarly journals Induced Coactions of Discrete Groups on C*-Algebras

1999 ◽  
Vol 51 (4) ◽  
pp. 745-770 ◽  
Author(s):  
Siegfried Echterhoff ◽  
John Quigg
Keyword(s):  
Quotient Group ◽  
Discrete Group ◽  
Discrete Groups ◽  
Crossed Products ◽  
C Algebras ◽  
Morita Equivalent ◽  
Close Relationship

AbstractUsing the close relationship between coactions of discrete groups and Fell bundles, we introduce a procedure for inducing a C*-coaction δ: D → D ⊗C*(G/N) of a quotient group G/N of a discrete group G to a C*-coaction Ind δ: Ind D → D ⊗C*(G) of G. We show that induced coactions behave in many respects similarly to induced actions. In particular, as an analogue of the well known imprimitivity theorem for induced actions we prove that the crossed products Ind D ×IndδG and D ×δG/N are always Morita equivalent. We also obtain nonabelian analogues of a theorem of Olesen and Pedersen which show that there is a duality between induced coactions and twisted actions in the sense of Green. We further investigate amenability of Fell bundles corresponding to induced coactions.

A CONTINUUM OF DISCRETE GROUP ACTIONS ON A FERMIONIC FACTOR

1993 ◽  
Vol 44 (3) ◽  
pp. 339-343 ◽  
Author(s):  
KAZUYUKI SAITÔ ◽  
J. D. MAITLAND WRIGHT
Keyword(s):  
Discrete Group ◽  
Group Actions

scholarly journals Positive Herz–Schur multipliers and approximation properties of crossed products

2017 ◽  
Vol 165 (3) ◽  
pp. 511-532 ◽  
Author(s):  
ANDREW MCKEE ◽  
ADAM SKALSKI ◽  
IVAN G. TODOROV ◽  
LYUDMILA TUROWSKA
Keyword(s):  
Discrete Group ◽  
Crossed Products ◽  
Approximation Properties ◽  
Completely Positive ◽  
Haagerup Property ◽  
Schur Multipliers

For aC*-algebraAand a setXwe give a Stinespring-type characterisation of the completely positive SchurA-multipliers on κ(ℓ2(X)) ⊗A. We then relate them to completely positive Herz–Schur multipliers onC*-algebraic crossed products of the formA⋊α,rG, withGa discrete group, whose various versions were considered earlier by Anantharaman-Delaroche, Bédos and Conti, and Dong and Ruan. The latter maps are shown to implement approximation properties, such as nuclearity or the Haagerup property, forA⋊α,rG.

A CONSTRUCTION FOR WEIGHTS ON C*-ALGEBRAS: DUAL WEIGHTS FOR C*-CROSSED PRODUCTS

1999 ◽  
Vol 10 (01) ◽  
pp. 129-157 ◽  
Author(s):  
J. QUAEGEBEUR ◽  
J. VERDING
Keyword(s):  
Dynamical System ◽  
Haar Measure ◽  
Crossed Product ◽  
Crossed Products ◽  
C Algebras ◽  
System A ◽  
Lower Semi ◽  
Continuous Weight ◽  
Densely Defined

A method for constructing densely defined lower semi-continuous weights on C*-algebras is presented. The method can be used to construct a "dual weight" on the C*-crossed product A×αG of a C*-dynamical system (A,G,α), starting from a relatively α-invariant densely defined lower semi-continuous weight on A. As an application we show that the Haar measure on the quantum E(2) group is a C*-dual weight.

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