Invitation to C ∗ -Algebras and Topological Dynamics

Tomiyama, Jun

doi:10.1142/0430

Cited by 55 publications

(38 citation statements)

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“…In what follows we restrict ourselves to two specific examples of Σ = (X, σ); Bernoulli shifts and irrational rotations. If Σ is a Bernoulli shift then C * (Σ) is not simple but primitive and if Σ is an irrational rotation on X = T then C * (Σ) is simple (see [6,8]). Among many ergodic measures on X we shall choose specific invariant probability measures on X and discuss ergodic extensions.…”

Section: Ergodic Extensionsmentioning

confidence: 99%

Topologically irreducible representations of the Banach -algebra associated with a dynamical system

Kishimoto¹,

Tomiyama²

2017

Ergod. Th. Dynam. Sys.

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We describe (infinite-dimensional) irreducible representations of the crossed product C * -algebra associated with a topological dynamical system (based on Z) and we show that their restrictions to the underling ℓ 1 -Banach * -algebra are not algebraically irreducible under mild conditions on the dynamical system. The above description of irreducible representations has two ingredients, ergodic measures on the space and ergodic extensions for the tensor product with type I factors; the latter which may not have been explicitly taken up before will be explored by examples. A new class of ergodic measures is also constructed for irrational rotations on the circle.

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Section: Ergodic Extensionsmentioning

confidence: 99%

Topologically irreducible representations of the Banach -algebra associated with a dynamical system

Kishimoto¹,

Tomiyama²

2017

Ergod. Th. Dynam. Sys.

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“…The interplay between dynamical and systems and operator theory is now a well developed subject [Tom92,Fur99,BJ91,Con94]. In particular, the operator theoretic approach to wavelet theory has been extremely productive [DL98, HL00, BJ99, BEJ00].…”

Section: Introductionmentioning

confidence: 99%

Irreducible wavelet representations and ergodic automorphisms on solenoids

Dutkay¹,

Larson²,

Silvestrov³

2011

Operators and Matrices

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Abstract. We focus on the irreducibility of wavelet representations. We present some connections between the following notions: covariant wavelet representations, ergodic shifts on solenoids, fixed points of transfer (Ruelle) operators and solutions of refinement equations. We investigate the irreducibility of the wavelet representations, in particular the representation associated to the Cantor set, introduced in [DJ06a], and we present several equivalent formulations of the problem.

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“…In [15] and [16], for example, one starts with a homeomorphism σ of a compact Hausdorff space X and constructs the crossed product C * -algebra C(X) ⋊ α Z, where C(X) is the algebra of continuous complex valued functions on X and α is the Z-action on C(X) naturally induced by σ. One of many results obtained is equivalence between simplicity of the algebra and minimality of the system, provided that X consists of infinitely many points, see [2], [9], [15], [16] or, for a more general approach in the metrizable case, [17]. In [12], a purely algebraic variant of the crossed product is considered, having more general classes of algebras than merely continuous functions on compact Hausdorff spaces as coefficient algebras.…”

Section: Introductionmentioning

confidence: 99%

Dynamical Systems Associated with Crossed Products

2009

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Abstract. In this paper, we consider both algebraic crossed products of commutative complex algebras A with the integers under an automorphism of A, and Banach algebra crossed products of commutative C * -algebras A with the integers under an automorphism of A. We investigate, in particular, connections between algebraic properties of these crossed products and topological properties of naturally associated dynamical systems. For example, we draw conclusions about the ideal structure of the crossed product by investigating the dynamics of such a system. To begin with, we recall results in this direction in the context of an algebraic crossed product and give simplified proofs of generalizations of some of these results. We also investigate new questions, for example about ideal intersection properties of algebras properly between the coefficient algebra A and its commutant A ′ . Furthermore, we introduce a Banach algebra crossed product and study the relation between the structure of this algebra and the topological dynamics of a naturally associated system.

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Invitation to C ∗ -Algebras and Topological Dynamics

Cited by 55 publications

References 0 publications

Topologically irreducible representations of the Banach -algebra associated with a dynamical system

Topologically irreducible representations of the Banach -algebra associated with a dynamical system

Irreducible wavelet representations and ergodic automorphisms on solenoids

Dynamical Systems Associated with Crossed Products

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