Type I orbits in the pure states of a \(C^ *\)-dynamical system
type: Item
instance of: scholarly article
title: Type I orbits in the pure states of a \(C^ *\)-dynamical system
zbMATH Open document ID: 0633.46062
author: Akitaka Kishimoto
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1987-01-01T00:00:00Z
review text: Groups of automorphisms on \(C^*\) algebras are considered: \(\alpha\) : \(G\to Aut(A)\). For such an action \(\alpha\) and a subgroup H of G, define the fixed-point-algebras, \(A^ G=\{a\in A:\alpha_ g(a)=a\), \(g\in G\}\) and \(A^ H=\{a\in A:\alpha_ h(a)=a\), \(h\in H\}\). For states f, let \(\pi_ f\) denote the GNS representation. For states f on the crossed product \(A\ltimes G\), the restriction \(\pi\) of \(\pi_ f\) to A is considered, and \(\rho =\int^{\oplus}_{G}\pi \circ \alpha_ tdt\). The relative commutant of \(\rho\) (A) is denoted \(\rho\) (A)'. Theorem 3.5 is reproduced below: Let A be a separable prime \(C^*\)-algebra and \(\alpha\) a faithful continuous action of a (separable) compact abelian group G on A. Let H be an arbitrary closed subgroup of G. Then the following conditions are equivalent: 1. \(A^ G\) is prime and there exists a G-invariant pure state f of A such that \(\pi_ f\) is faithful. 2. \(A^ H\) is prime and there exists an H-invariant pure state \(\phi\) of A such that \(\pi_{\phi}\) is faithful and \(\rho_{\phi}(A)''\cap \rho_{\phi}(A)'=\{p:\) \(p\in H^{\perp}\}''\otimes 1\), where \(A^ H=A^{\alpha | H}(0)\), etc.
Mathematics Subject Classification ID: 46L55
Mathematics Subject Classification ID: 46L30
Mathematics Subject Classification ID: 46L40
zbMATH DE Number: 4030269
zbMATH Keywords: locally compact abelian group
zbMATH Keywords: crossed product
zbMATH Keywords: prime \(C^ *\)-algebra
zbMATH Keywords: Groups of automorphisms on \(C^ *\) algebras
zbMATH Keywords: GNS representation
zbMATH Keywords: relative commutant
zbMATH Keywords: faithful continuous action
zbMATH Keywords: invariant pure state
reviewed by: Palle Jorgensen
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195176543
OpenAlex ID: W1482098502
cites work: Quasi-product actions of a compact abelian group on a \(C^*\)-algebra
cites work: Q3730466
cites work: Embedding product type actions into C *-dynamical systems
cites work: Outer conjugacy classes of automorphisms of factors
cites work: Outer automorphisms and reduced crossed products of simple C*-algebras
cites work: Q3687242
cites work: Q3205815
cites work: Q3876492
DOI: 10.2977/PRIMS/1195176543
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description: scientific article; zbMATH DE number 4030269