Invariant means and finite representation theory of C*-algebras

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^ω-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.

This is the final revision! Really. The paper was accepted by Memoirs of AMS, so most of the changes amount to adding details and discussion. 104 pages

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