Weak expectations in \(C^*\)-dynamical systems
type: Item
instance of: scholarly article
title: Weak expectations in \(C^*\)-dynamical systems
zbMATH Open document ID: 0672.46040
author: Charles J. K. Batty
author: Masaharu Kusuda
published in: Journal of the Mathematical Society of Japan
publication date: 1988-01-01T00:00:00Z
review text: \((A,G,\alpha)\) being a \(C^*\)-dynamical system and B an \(\alpha\)- invariant \(C^*\)-subalgebra of A, let \(({\mathcal H},\pi,u)\) be a covariant representation of \((B,G,\alpha)\) and \(M=(\pi(B)\cup u(G))''\). A covariant weak expectation is a completely positive linear contraction \(Q: A\to M\) such that \(Q| B=\pi\) and \(Q(\alpha_ t(a))=u_ tQ(a)u^*_ t\) where \(u\in A\) and \(t\in G\). Its existence is shown to be equivalent to the existence of a weak expectation from the \(C^*\)-crossed product \(A_{\alpha}\times G\) into M, i.e. a linear contraction \(\hat Q\) such that \(\hat Q(y)=(\pi \times u)\), \(y\in L^ 1(G,B)\). Analogous results are obtained for covariant representations associated to an \(\alpha\)- invariant state \(\phi\) of B. Moreover the existence of Q or \(\hat Q\) is shown to be asserted if \(\phi\) is G-central.
reviewed by: G. Loupias
Mathematics Subject Classification ID: 46L55
zbMATH DE Number: 4099953
zbMATH Keywords: \(C^*\)-dynamical system
zbMATH Keywords: covariant representation
zbMATH Keywords: covariant weak expectation
zbMATH Keywords: completely positive linear contraction
zbMATH Keywords: \(C^*\)-crossed product
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/04040661
OpenAlex ID: W2011111488
DOI: 10.2969/JMSJ/04040661
description: scientific article; zbMATH DE number 4099953