The G-central decomposition of states of statistical systems in the algebraic and in the operational description
type: Item
instance of: scholarly article
title: The G-central decomposition of states of statistical systems in the algebraic and in the operational description
zbMATH Open document ID: 0589.46046
DOI: 10.1016/0034-4877(85)90034-5
author: Maria Cristina Abbati
author: Alessandro Manià
published in: Reports on Mathematical Physics
publication date: 1985-01-01T00:00:00Z
review text: Let X be a compact convex set, regularly embedded in a locally convex space E. The space E is ordered by the cone \(E^+\) with base X. Let g be a group of continuous order-automorphisms of E, mapping X onto itself. It is shown that for each point x of X there exists a unique G-central measure for x, and that the measure is pseudoconcentrated on the set of all G-primary points of X. This generalizes to compact convex sets the construction carried out by \textit{A. Guichardet} and \textit{D. Kastler} in J. Math. Pures Appl., IX. Ser. 49, 349-380 (1970; Zbl 0185.387) who considered the action of a group G of automorphisms of a \(C^*\)-algebra A and for each quasi-invariant state x found a \(''Z^ 0\)-central'' measure representing x, pseudoconcentrated on the set of \(''Z^ 0\)-pure'' states of A \((Z^ 0\) being the set of all G-invariant elements in the center of \(A^{**})\). The authors' notions of G-central measure and G- primary points correspond exactly to the notions of \(Z^ 0\)-central measure and \(Z^ 0\)-pure states if X is the set of states of a \(C^*\)- algebra A. The main result of Guichardet and Kastler, with the separability restriction removed, is obtained as an application.
Mathematics Subject Classification ID: 46L45
Mathematics Subject Classification ID: 46L60
Mathematics Subject Classification ID: 46A03
zbMATH DE Number: 3946921
zbMATH Keywords: compact convex set
zbMATH Keywords: central decomposition
zbMATH Keywords: direct integral
zbMATH Keywords: group of continuous order-automorphisms
zbMATH Keywords: central measure
zbMATH Keywords: pseudoconcentrated
zbMATH Keywords: primary points
zbMATH Keywords: quasi-invariant state
reviewed by: Stanisław Goldstein
MaRDI profile type: Publication
full work available at URL: 0034-4877(85)90034-5
OpenAlex ID: W2027105834
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description: scientific article; zbMATH DE number 3946921