Properties of Topological Dynamical Systems and Corresponding $C^*$-Algebras

Kawamura, Shinzô; Tomiyama, Jun

doi:10.3836/tjm/1270132260

Cited by 38 publications

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“…The following corollary is a more accurate statement than Theorem 4.4 of [7]. In [9] the term regular was introduced to describe a C*-dynamical system for which the full…”

Section: Remarkmentioning

confidence: 99%

“…The implication (i)^-(ii) follows from Theorem 1. The proof of (ii)=>(i) essentially appears in [7]. For completeness we present the following brief proof, which is more self-contained than that in [7].…”

Section: Proof Let T U T N Eg\{e}mentioning

confidence: 99%

“…A condition on the ideals of the reduced crossed product is presented which corresponds to the dynamical condition of proper outerness. This is used to give another proof of simplicity of reduced crossed-products.In this paper we present a somewhat simpler proof than that of [7] of the equivalence of these two conditions. In one direction, that the dynamical condition implies the condition on ideals, the argument is valid for arbitrary discrete systems.…”

mentioning

confidence: 99%

“…In this paper we present a somewhat simpler proof than that of [7] of the equivalence of these two conditions. In one direction, that the dynamical condition implies the condition on ideals, the argument is valid for arbitrary discrete systems.…”

mentioning

confidence: 99%

“…In [8] a general version is proved using a condition (apparently) stronger than proper outerness. In [7] the discrete abelian case is studied. A condition on the ideals of the reduced crossed product is presented which corresponds to the dynamical condition of proper outerness.…”

mentioning

confidence: 99%

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Topologically free actions and ideals in discrete C*-dynamical systems

Archbold

Spielberg

1994

Proceedings of the Edinburgh Mathematical Society

107

155

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A C'-dynamical system is called topologically free if the action satisfies a certain natural condition weaker than freeness. It is shown that if a discrete system is topologically free then the ideal structure of the crossed product algebra is related to that of the original algebra. One consequence is that a minimal topologically free discrete system has a simple reduced crossed product. Sharper results are obtained when the algebra is abelian.1991 Mathematics subject classification: Primary 46L55.Let (A, G,a) be a C*-dynamical system with A abelian and G discrete. In [4] it was shown that the reduced crossed-product A x ^G is simple if G acts minimally and if each automorphism a,, t^e, is properly outer (see below for the definition). As a minimal action can be properly outer without being free, this was a significant advance from earlier results of Zeller-Meier and Effros-Hahn (see [9, Corollary 4.5] for naturally occurring examples). Elliott's result applies to actions of discrete groups on a larger class of C*-algebras than abelian ones. In [8] a general version is proved using a condition (apparently) stronger than proper outerness. In [7] the discrete abelian case is studied. A condition on the ideals of the reduced crossed product is presented which corresponds to the dynamical condition of proper outerness. This is used to give another proof of simplicity of reduced crossed-products.In this paper we present a somewhat simpler proof than that of [7] of the equivalence of these two conditions. In one direction, that the dynamical condition implies the condition on ideals, the argument is valid for arbitrary discrete systems. In fact, this implication is contained in the proof of [4] and [8] (using the apparently stronger condition of [8]); however, in the general case our argument is much simpler. We remark that no assumptions are made regarding separability, countability, or possession of an identity element. All ideal are assumed to be closed and two-sided.Much of the work described in this article was done while the second author was an SERC fellow at the University of Aberdeen. He wishes to thank SERC for its support, and the members of the Mathematics Department at the University of Aberdeen for their warm hospitality.

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“…The following corollary is a more accurate statement than Theorem 4.4 of [7]. In [9] the term regular was introduced to describe a C*-dynamical system for which the full…”

Section: Remarkmentioning

confidence: 99%

Section: Proof Let T U T N Eg\{e}mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Topologically free actions and ideals in discrete C*-dynamical systems

Archbold

Spielberg

1994

Proceedings of the Edinburgh Mathematical Society

107

155

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Dynamics, Wavelets, Commutants and Transfer Operators Satisfying Crossed Product Type Commutation Relations

Silvestrov

2013

Springer Proceedings in Mathematics &Amp; Statistics

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Regular Ideals, Ideal Intersections, and Quotients

Brown,

Fuller,

Pitts

et al. 2024

Integr. Equ. Oper. Theory

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Let $$B \subseteq A$$ B ⊆ A be an inclusion of $$C^*$$ C ∗ -algebras. We study the relationship between the regular ideals of B and regular ideals of A. We show that if $$B \subseteq A$$ B ⊆ A is a regular $$C^*$$ C ∗ -inclusion and there is a faithful invariant conditional expectation from A onto B, then there is an isomorphism between the lattice of regular ideals of A and invariant regular ideals of B. We study properties of inclusions preserved under quotients by regular ideals. This includes showing that if $$D \subseteq A$$ D ⊆ A is a Cartan inclusion and J is a regular ideal in A, then $$D/(J\cap D)$$ D / ( J ∩ D ) is a Cartan subalgebra of A/J. We provide a description of regular ideals in the reduced crossed product of a C$$^*$$ ∗ -algebra by a discrete group.

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Properties of Topological Dynamical Systems and Corresponding $C^*$-Algebras

Cited by 38 publications

References 0 publications

Topologically free actions and ideals in discrete C*-dynamical systems

Topologically free actions and ideals in discrete C*-dynamical systems

Dynamics, Wavelets, Commutants and Transfer Operators Satisfying Crossed Product Type Commutation Relations

Regular Ideals, Ideal Intersections, and Quotients

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