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 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.

Morita equivalences between fixed point algebras and crossed products

1999 ◽  
Vol 125 (1) ◽  
pp. 43-52 ◽  
Author(s):  
CHI-KEUNG NG
Keyword(s):  
Fixed Point ◽  
Quantum Group ◽  
Type Theorem ◽  
Compact Quantum Group ◽  
Crossed Product ◽  
Crossed Products ◽  
Morita Equivalent ◽  
Fixed Point Algebra

In this paper, we will prove that if A is a C*-algebra with an effective coaction ε by a compact quantum group, then the fixed point algebra and the reduced crossed product are Morita equivalent. As an application, we prove an imprimitivity type theorem for crossed products of coactions by discrete Kac C*-algebras.

scholarly journals Almost Finiteness for General Étale Groupoids and Its Applications to Stable Rank of Crossed Products

2018 ◽  
Vol 2020 (19) ◽  
pp. 6007-6041 ◽  
Author(s):  
Yuhei Suzuki
Keyword(s):  
Amenable Group ◽  
Cantor Set ◽  
Stable Rank ◽  
Crossed Product ◽  
Local Version ◽  
Crossed Products ◽  
Subexponential Growth ◽  
Minimal Action ◽  
New Strategy ◽  
Totally Disconnected

Abstract We extend Matui’s notion of almost finiteness to general étale groupoids and show that the reduced groupoid C$^{\ast }$-algebras of minimal almost finite groupoids have stable rank 1. The proof follows a new strategy, which can be regarded as a local version of the large subalgebra argument. The following three are the main consequences of our result: (1) for any group of (local) subexponential growth and for any its minimal action admitting a totally disconnected free factor, the crossed product has stable rank 1; (2) any countable amenable group admits a minimal action on the Cantor set, all whose minimal extensions form the crossed product of stable rank 1; and (3) for any amenable group, the crossed product of the universal minimal action has stable rank 1.

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.

Crossed Products by Semigroups of Endomorphisms and Groups of Partial Automorphisms

2003 ◽  
Vol 46 (1) ◽  
pp. 98-112 ◽  
Author(s):  
Nadia S. Larsen
Keyword(s):  
Fixed Point ◽  
Dynamical Systems ◽  
Crossed Product ◽  
Crossed Products ◽  
Partial Action ◽  
Multiplier Algebra ◽  
Fixed Point Algebra

AbstractWe consider a class (A; S; α) of dynamical systems, where S is an Ore semigroup and α is an action such that each αs is injective and extendible (i.e. it extends to a non-unital endomorphism of the multiplier algebra), and has range an ideal of A. We show that there is a partial action on the fixed-point algebra under the canonical coaction of the enveloping group G of S constructed in [15, Proposition 6.1]. It turns out that the full crossed product by this coaction is isomorphic to A ⋊αS. If the coaction is moreover normal, then the isomorphism can be extended to include the reduced crossed product. We look then at invariant ideals and finally, at examples of systems where our results apply.

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.

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.

scholarly journals C*-Crossed-Products by an Order-Two Automorphism

2010 ◽  
Vol 53 (1) ◽  
pp. 37-50
Author(s):  
Man-Duen Choi ◽  
Frédéric Latrémolière
Keyword(s):  
Fixed Point ◽  
Representation Theory ◽  
Free Group ◽  
Cyclic Group ◽  
Crossed Product ◽  
Irreducible Representations ◽  
Crossed Products

AbstractWe describe the representation theory of C*-crossed-products of a unital C*-algebra A by the cyclic group of order 2. We prove that there are two main types of irreducible representations for the crossed-product: those whose restriction to A is irreducible and those who are the sum of two unitarily unequivalent representations of A. We characterize each class in term of the restriction of the representations to the fixed point C*-subalgebra of A. We apply our results to compute the K-theory of several crossed-products of the free group on two generators.

scholarly journals The Dixmier-Douady class of groupoid crossed products

2004 ◽  
Vol 76 (2) ◽  
pp. 223-234 ◽  
Author(s):  
Paul S. Muhly ◽  
Dana P. Williams
Keyword(s):  
Crossed Product ◽  
Crossed Products ◽  
Continuous Trace

AbstractWe give a formula for the Dixmier-Douady class of a continuous-trace groupoid crossed product that arises from an action of a locally trivial, proper, principal groupoid on a bundle of elementary C*-algebras that satisfies Fell's condition.

scholarly journals A rigidity result for crossed products of actions of Baumslag–Solitar groups

2015 ◽  
Vol 26 (14) ◽  
pp. 1550117
Author(s):  
Niels Meesschaert
Keyword(s):  
Probability Measure ◽  
Free Probability ◽  
Crossed Product ◽  
Crossed Products ◽  
Orbit Equivalence ◽  
Rigidity Result ◽  
Profinite Completions ◽  
Abelian Subgroups ◽  
Measure Equivalence ◽  
Measure Preserving Actions

Let [Formula: see text] and [Formula: see text] be two ergodic essentially free probability measure preserving actions of nonamenable Baumslag–Solitar groups whose canonical almost normal abelian subgroups act aperiodically. We prove that an isomorphism between the corresponding crossed product II1 factors forces [Formula: see text] when [Formula: see text] and [Formula: see text] when [Formula: see text]. This improves an orbit equivalence rigidity result obtained by Houdayer and Raum in [Baumslag–Solitar groups, relative profinite completions and measure equivalence rigidity, J. Topol. 8 (2015) 295–313].

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