The G-central decomposition of states of statistical systems in the algebraic and in the operational description
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.
scientific article; zbMATH DE number 3946921
scientific article