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. 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.