Pseudocompact C$^*$-algebras

TL;DR

This paper investigates pseudocompact C*-algebras, logical limits of finite-dimensional algebras, detailing their properties, structure, and subclasses, including their K-theoretic and trace characteristics.

Contribution

It provides a comprehensive analysis of pseudocompact C*-algebras, including their axiomatization, stability properties, and the structure of their subclasses like pseudomatricial algebras.

Findings

Pseudocompact C*-algebras are unital, stably finite, and have real rank zero.

They have trivial K_1 groups and the Dixmier property.

The subclass of pseudomatricial C*-algebras has unique traces and trivial centers.

Abstract

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.