Invariant means and finite representation theory of C*-algebras

TL;DR

This paper explores invariant means and the structure of II_1-factor representations in unital C*-algebras, providing new insights into their classification, embeddability, and applications to operator algebra problems.

Contribution

It introduces a general theory linking invariant means to finite representation theory and applies it to classify certain C*-algebras and analyze their representations.

Findings

Characterization of R^{}-embeddable factors via WEP

Classification of simple, nuclear C*-algebras with unique trace

Existence of filtrations for finite section methods in operators

Abstract

Various subsets of the tracial state space of a unital C*-algebra are studied. The largest of these subsets has a natural interpretation as the space of invariant means. II_1-factor representations of a class of C*-algebras considered by Sorin Popa are also studied. These algebras are shown to have an unexpected variety of II_1-factor representations. This general theory is related to various other problems as well. Applications include: (1) A characterization of R^{\omega}-embeddable factors in terms of Lance's WEP. (2) A classification theorem for certain simple, nuclear C*-algebras with unique trace. (3) For a self-adjoint operator there always exists a filtration such that the finite section method (from numerical analysis) works as well as could be hoped for. (4) New examples of non-tracially AF algebras which answer negatively questions of Sorin Popa and Huaxin Lin.