Quasitraces on exact C*-algebras are traces

TL;DR

This paper proves that all 2-quasitraces on unital exact C*-algebras are actually traces, leading to significant implications for the structure of such algebras and their relation to von Neumann factors.

Contribution

It establishes that 2-quasitraces on unital exact C*-algebras are traces, resolving a longstanding problem and impacting the understanding of algebraic structures.

Findings

All 2-quasitraces on unital exact C*-algebras are traces

Every stably finite exact unital C*-algebra has a tracial state

AW*-factors generated by exact C*-subalgebras are von Neumann II_1-factors

Abstract

It is shown that all 2-quasitraces on a unital exact C*-algebra are traces. As consequences one gets: (1) Every stably finite exact unital C*-algebra has a tracial state, and (2) if an AW*-factor of type II_1 is generated (as an AW*-algebra) by an exact C*-subalgebra, then it is a von Neumann II_1-factor. This is a partial solution to a well known problem of Kaplansky. The present result was used by Blackadar, Kumjian and R{\o}rdam to prove that RR(A)=0 for every simple non-commutative torus of any dimension.