Non-unital tracially approximated ${\rm C^*}$-algebras

TL;DR

This paper introduces a new class of non-unital tracially approximated C*-algebras and shows they inherit tracial $ ext{Z}$-absorption properties from related classes, advancing understanding of their structural properties.

Contribution

The paper defines non-unital tracial approximation C*-algebras and proves they are tracially $ ext{Z}$-absorbing if related classes are.

Findings

Non-unital tracial approximation C*-algebras are introduced.

Such algebras inherit tracial $ ext{Z}$-absorption from related classes.

The results unify and extend existing concepts in the structure theory of C*-algebras.

Abstract

In this paper, we introduce a class of non-unital tracial approximation ${\rm C^*}$-algebras. Consider the class of ${\rm C^*}$-algebras which are tracially $\mathcal{Z}$-absorbing (in the sense of Amint, Golestani, Jamali, Phillips's simple tracially $\mathcal{Z}$-absorbing or Castillejos, Li, Szabvo's tracial $\mathcal{Z}$-stability). Then $A$ is tracially $\mathcal{Z}$-absorbing for any simple ${\rm C^*}$-algebra $A$ in the corresponding class of non-unital tracial approximation ${\rm C^*}$-algebras.