Diffuse traces and Haar unitaries

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 the situation for nontracial states. 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.

Nowhere scattered C*-algebras

“…If A is unital, then a unitary u ∈ A is said to be a Haar unitary with respect to ϕ if ϕ(u k ) = 0 for all k ∈ Z \ {0}; see [Thi20a,Definition 4.8]. This definition is a generalization to the setting of positive functionals of the well-established notion of Haar unitaries with respect to traces.…”

“…The term 'nowhere scattered' was first used in [Thi20a] to name positive functionals on C * -algebras that give no weight to scattered ideal-quotients. It is the purpose of this paper to initiate a systematic study of nowhere scatteredness by collecting, systematizing and complementing existing results.…”