Structure of ideals and isomorphisms of \(C^*\)-crossed products by single homeomorphism
type: Item
instance of: scholarly article
title: Structure of ideals and isomorphisms of \(C^*\)-crossed products by single homeomorphism
zbMATH Open document ID: 0979.46046
author: Jun Tomiyama
published in: Tokyo Journal of Mathematics
publication date: 2002-01-30T00:00:00Z
review text: The homeomorphism \(C^*\)-algebra for a topological dynamical system \(\Sigma=(X,\sigma)\), where \(X\) is a compact Hausdorff space (in general, non-metrizable) and \(\sigma\) its homeomorphism, is the crossed product \(A(\Sigma)=C(X)\times_\alpha{\mathbb Z}\) with respect to the action of \({\mathbb Z}\) by powers of \(\alpha\in \Aut(C(X))\), given by \((\alpha f)(x)=f(\sigma^{-1}x)\) [see the author, ``The interplay between topological dynamics and theory of \(C^*\)-algebras'' (1992; Zbl 0892.46070) and \textit{M. Boyle} and the author, J. Math. Soc. Japan 50, No. 2, 317-329 (1998; Zbl 0940.37004)]. The paper studies the ideals and isomorphisms of \(A(\Sigma)\). At first Sect. 2 developes an approximation techniques, involving the convolution with summability kernels on the torus \(T\), for natural Fourier series expansion of the elements of the ``universal'' \(C^*\)-crossed product by \({\mathbb Z}\). Sect. 3 describes the ideals in terms of the kernels of irreducible representations of \(A(\Sigma)\) associated to periodic and aperiodic points of \(\sigma\). A class of ``well-behaved'' ideals, including the kernels of infinite dimensional irreducible representations, is investigated. In Sect. 4 the isomorphisms between \(A(\Sigma_1)\) and \(A(\Sigma_2)\) preserving the mentioned class of ideals are called standard. If the set of periodic points of \(\sigma\) is at most countable every isomorphism through \(A(\Sigma)\) is standard. A general notion of algebraic invariants of \(\Sigma\), i.e., the properties shared by systems with isomorphic \(A(\Sigma)\), is introduced and illustrated (e.g., minimality, topological transitivity, freeness).
Mathematics Subject Classification ID: 46L55
Mathematics Subject Classification ID: 37B20
zbMATH DE Number: 1523366
zbMATH Keywords: topological dynamical system
zbMATH Keywords: homeomorphism \(C^*\)-algebra
zbMATH Keywords: crossed product by \({\mathbb Z}\)
zbMATH Keywords: ideal
zbMATH Keywords: isomorphisms
zbMATH Keywords: approximation techniques
zbMATH Keywords: convolution with summability kernels
zbMATH Keywords: natural Fourier series expansion
zbMATH Keywords: irreducible representations
zbMATH Keywords: algebraic invariants
zbMATH Keywords: minimality
zbMATH Keywords: topological transitivity
zbMATH Keywords: freeness
reviewed by: Andrey V. Bulinski
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1255958804
OpenAlex ID: W1972586920
DOI: 10.3836/TJM/1255958804
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description: scientific article; zbMATH DE number 1523366