Type I orbits in the pure states of a \(C^ *\)-dynamical system. II
type: Item
instance of: scholarly article
title: Type I orbits in the pure states of a \(C^ *\)-dynamical system. II
zbMATH Open document ID: 0642.46057
author: Akitaka Kishimoto
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1987-01-01T00:00:00Z
review text: For a \(C^*\)-algebra A with an action \(\alpha\) of a locally compact abelian group G, one considers the pure states of A, or rather the spectrum of A with the associated action. Type I orbits are defined and studied in the previous paper [ibid. 23, 321-336 (1987; Zbl 0633.46062)]. We continue this study and show that if A is separable and simple and G is separable and if there is a type I orbit through a pure state f with trivial stabilizer \(\{t\in G:\pi_ f\circ \alpha_ t\sim \pi_ f\}=\{0\}\), then there is a type I orbit with stabilizer equal to any given closed subgroup of G. This in particular shows that if the system is asymptotically abelian in the sense that there is an automorphism \(\gamma\) of A such hat \(\gamma \circ \alpha_ t=\alpha_ 1\circ \gamma\), \(t\in G\), and \([\gamma^ n(x),y]\to 0\) as \(n\to \infty\) for any x,y\(\in A\), there is an \(\alpha\)-covariant irreducible representation.
Mathematics Subject Classification ID: 46L55
Mathematics Subject Classification ID: 46L40
Mathematics Subject Classification ID: 46L30
zbMATH DE Number: 4046426
zbMATH Keywords: C\({}^ *\)-dynamical system
zbMATH Keywords: Connes spectrum
zbMATH Keywords: crossed products
zbMATH Keywords: pure states
zbMATH Keywords: Type I orbits
zbMATH Keywords: trivial stabilizer
zbMATH Keywords: asymptotically abelian
zbMATH Keywords: covariant irreducible representation
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195176444
OpenAlex ID: W2806367376
cites work: Quasi-product actions of a compact abelian group on a \(C^*\)-algebra
cites work: Outer automorphisms and reduced crossed products of simple C*-algebras
cites work: Q3937869
cites work: Q3205815
DOI: 10.2977/PRIMS/1195176444
description: scientific article; zbMATH DE number 4046426