Invariant means and finite representation theory of C*-algebras

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. II₁-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 II₁-factor representations. In addition to developing some general theory we also show that these ideas are related to numerous other problems in operator algebras.