Choquet order and Jordan maps
This review can only be understood by recalling previous results by the same authors [Int. J. Theor. Phys. 55, No. 7, 3353--3365 (2016; Zbl 1358.81019)] where they have shown the existence of a connection between the Choquet order of orthogonal measures and posets of abelian subalgebras of von Neumann algebras. By employing the commented relationship they proved that any isomorphism between the ordered set of orthogonal measures with a fixed barycenter endowed with the Choquet order is induced by a Jordan \(^*\)-isomorphism of certain associated von Neumann algebras. The isomorphisms of the whole posets of orthogonal measures were treated by the same authors in [J. Math. Sci., New York 241, No.~4, 501--506 (2019; Zbl 1453.46053)]. In this interesting note the authors consider a more general situation in which states and their decompositions are invariant with respect to an action of a discrete automorphism group of a given C\(^*\)-algebra. In this situation, the Choquet order is identified with set theoretic order of abelian von Neumann subalgebras associated with covariant GNS representations. They also establish a discrete version of this result for orthogonal decompositions of invariant states. This paper is part of an interesting research line developed by the authors devoted to explore the interplay between two structures relevant to the foundations of quantum theory that have been investigated separately so far: the structure of decompositions of states on operator algebras (which has important applications to quantum statistical mechanics, to describing equilibria of quantum systems, and to the convex approach to quantum theory in general), and the ``context structure'' of abelian subalgebras of an operator algebra \(M\) ordered by set theoretic inclusion.
scientific article; zbMATH DE number 6909456
scientific article