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.
Publication Year: 2020
Publication Date: 2020-09-15
Language: en
Type: preprint
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