A tracially AF algebra which is not $\mathcal Z$-absorbing

TL;DR

This paper constructs a specific simple, separable, unital, non-nuclear tracially AF algebra that does not tensorially absorb the Jiang-Su algebra, challenging assumptions about their relationship.

Contribution

It provides a counterexample of a tracially AF algebra that is not $ ext{Z}$-absorbing, highlighting limitations in the classification of such algebras.

Findings

Existence of a non-$ ext{Z}$-absorbing tracially AF algebra

Counterexample to the conjecture that all tracially AF algebras absorb $ ext{Z}$

Implications for classification theory of C*-algebras

Abstract

We show that there is a simple separable unital (non-nuclear) tracially AF algebra $A$ which does not absorb the Jiang-Su algebra $\mathcal Z$ tensorially, i.e., $A \ncong A\otimes\mathcal Z$.