The Blackadar-Handelman theorem for non-unital C*-algebras

TL;DR

This paper investigates whether the Blackadar-Handelman theorem, linking stable finiteness and quasitraces, extends to non-unital C*-algebras, providing positive results for well-behaved cases and counterexamples in general.

Contribution

It demonstrates that the Blackadar-Handelman theorem does not fully extend to non-unital C*-algebras by constructing specific counterexamples.

Findings

Counterexample of a non-unital C*-algebra with finite multiplier algebras but no bounded quasitrace

Counterexample of a simple, stably non-stable non-unital C*-algebra without a bounded quasitrace

Positive results for well-behaved non-unital C*-algebras regarding the theorem's extension

Abstract

A well-known theorem of Blackadar and Handelman states that every unital stably finite C*-algebra has a bounded quasitrace. Rather strong generalizations of stable finiteness to the non-unital case can be obtained by either requiring the multiplier algebra to be stably finite, or alternatively requiring it to be at least stably not properly infinite. This paper deals with the question whether the Blackadar-Handelman result can be extended to the non-unital case with respect to these generalizations of stable finiteness. For suitably well-behaved C*-algebras there is a positive result, but none of the non-unital versions holds in full generality. Two examples of C*-algebras are constructed. The first one is a non-unital, stably commutative C*-algebra $A$ that contradicts the weakest possible generalization of the Blackadar-Handelman theorem: The multiplier algebras of all matrix algebras over $A$ are finite, while $A$ has no bounded quasitrace. The second example is a non-unital, simple C*-algebra $B$ that is stably non-stable, i.e. no matrix algebra over $B$ is a stable C*-algebra. In fact, the multiplier algebras over all matrix algebras of this C*-algebra are not properly infinite. Moreover, the C*-algebra $B$ has no bounded quasitrace and therefore gives a simple counterexample to a possible generalization of the Blackadar-Handelman theorem.