Ultraproducts for State-Spaces of C*-Algebra and Radon Measures

Abstract

© 2020, PleiadesT Publishing,T Ltd. Abstract—This paper deals with properties of the ultraproducts for various structures. We introduce and study the concept of the ergodic action of a group with respect to a normal state on an abelian von Neumann algebra. In particular, we provide an example showing that the ultraproduct of ergodic states, generally speaking, is not ergodic. The ultraproduct of the Radon measures on a compact convex subset of a locally convex space is also investigated in the paper. As is well-known, the study of the extreme points in the state set for a C*−algebra is a very interesting problem in itself. Considering the ultraproducts of C*-algebras and the states on these algebras, we get quite nontrivial results.