Pseudocompact C$^*$-algebras

We study the class of pseudocompact C*-algebras, which are the logical limits of finite-dimensional C*-algebras. The pseudocompact C*-algebras are unital, stably finite, real rank zero, stable rank one, and tracial. We show that the pseudocompact C*-algebras have trivial K_1 groups and the Dixmier property. The class is stable under direct sums, tensoring by finite-dimensional C*-algebras, taking corners, and taking centers. We give an explicit axiomatization of the commutative pseudocompact C*-algebras. We also study the subclass of pseudomatricial C*-algebras, which have unique tracial states, strict comparison of projections, and trivial centers. We give some information about the K_0 groups of the pseudomatricial C*-algebras.