Tracially sequentially-split ${}^*$-homomorphisms between $C^*$-algebras

TL;DR

This paper introduces a tracial analogue of sequentially split *-homomorphisms for $C^*$-algebras, showing how important properties pass between algebras and applying this to actions with the tracial Rokhlin property.

Contribution

It defines a tracial version of sequentially split *-homomorphisms and demonstrates their role in transferring properties and analyzing Rokhlin actions in $C^*$-algebra theory.

Findings

Tracial Rokhlin property induces tracial sequentially split *-homomorphisms.

Approximation properties pass from target to domain algebra via these maps.

Unified approach to permanence properties related to tracial Rokhlin property.

Abstract

We define a tracial analogue of the sequentially split $*$-homomorphism between $C^*$-algebras of Barlak and Szab\'{o} and show that several important approximation properties related to the classification theory of $C^*$-algebras pass from the target algebra to the domain algebra. Then we show that the tracial Rokhlin property of the finite group $G$ action on a $C^*$-algebra $A$ gives rise to a tracial version of sequentially split $*$-homomorphism from $A\rtimes_{\alpha}G$ to $M_{|G|}(A)$ and the tracial Rokhlin property of an inclusion $C^*$-algebras $A\subset P$ with a conditional expectation $E:A \to P$ of a finite Watatani index generates a tracial version of sequentially split map. By doing so, we provide a unified approach to permanence properties related to tracial Rokhlin property of operator algebras.