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. In addition to developing some general theory we also show that these ideas are related to numerous other problems in operator algebras.

Table of Contents

• Introduction
• Notation, definitions and useful facts
• Amenable traces and stronger approximation properties
• Examples and special cases
• Finite representations
• Applications and connections with other areas
• Bibliography