Diffuse traces and Haar unitaries

TL;DR

This paper characterizes when tracial states on unital C*-algebras admit Haar unitaries, linking diffuseness to the absence of finite-dimensional representations, and explores the existence of Haar unitaries for nontracial states, with applications to free product structures.

Contribution

It provides a complete characterization of when tracial states admit Haar unitaries and extends the analysis to nontracial states on simple, infinite-dimensional C*-algebras.

Findings

A tracial state admits a Haar unitary iff it is diffuse.

Unital C*-algebras with no finite-dimensional representations have all tracial states admitting Haar unitaries.

The tracial reduced free product of simple C*-algebras is simple with stable rank one.

Abstract

We show that a tracial state on a unital C*-algebra admits a Haar unitary if and only if it is diffuse, if and only if it does not dominate a tracial functional that factors through a finite-dimensional quotient. It follows that a unital C*-algebra has no finite-dimensional representations if and only if each of its tracial states admits a Haar unitary. More generally, we study when nontracial states admit Haar unitaries. In particular, we show that every state on a unital, simple, infinite-dimensional C*-algebra admits a Haar unitary. We obtain applications to the structure of reduced free products. Notably, the tracial reduced free product of simple C*-algebras is always a simple C*-algebra of stable rank one.