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.