• Corpus ID: 221703061

Diffuse traces and Haar unitaries

@article{Thiel2020DiffuseTA,
  title={Diffuse traces and Haar unitaries},
  author={Hannes Thiel},
  journal={arXiv: Operator Algebras},
  year={2020}
}

• Hannes Thiel
• Published 15 September 2020
• Mathematics
• arXiv: Operator Algebras

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.