Simple tracially $\mathcal{Z}$-absorbing C*-algebras

TL;DR

This paper introduces a new notion of tracial $\\mathcal{Z}$-absorption for simple C*-algebras, explores its properties, and provides examples including purely infinite cases that are not $\\mathcal{Z}$-absorbing, expanding understanding of these algebras.

Contribution

It extends the concept of tracial $\\mathcal{Z}$-absorption to non-unital algebras, studies its permanence, and constructs new examples using free product techniques.

Findings

Razak-Jacelon algebra is tracially $\mathcal{Z}$-absorbing

First purely infinite examples not $\mathcal{Z}$-absorbing found

Cuntz semigroup is almost unperforated and algebra is weakly almost divisible

Abstract

We define a notion of tracial $\mathcal{Z}$-absorption for simple not necessarily unital C*-algebras, study it systematically, and prove its permanence properties. This extends the notion defined by Hirshberg and Orovitz for unital C*-algebras. The Razak-Jacelon algebra, simple C*-algebras with tracial rank zero, and simple purely infinite C*-algebras are tracially $\mathcal{Z}$-absorbing. We obtain the first purely infinite examples of tracially $\mathcal{Z}$-absorbing C*-algebras which are not $\mathcal{Z}$-absorbing. We use techniques from reduced free products of von~Neumann algebras to construct these examples. A stably finite example was given by Z. Niu and Q. Wang in 2021. We study the Cuntz semigroup of a simple tracially $\mathcal{Z}$-absorbing C*-algebra and prove that it is almost unperforated and the algebra is weakly almost divisible.