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

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.Comment: A serious flaw in Definition 2.6 has been notified to the authors. We fix our definition and accordingly change statements in subsequent propositions and theorems. Moreover, a gap in the proof of Theorem 2.25 is fixed. We note our appreciation for such helpful comments in Acknowledgements section. Some typos are also caught. We hope that it is fina