We define a tracial analogue of the sequentially split β-homomorphism
between Cβ-algebras of Barlak and Szab\'{o} and show that several important
approximation properties related to the classification theory of Cβ-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 Cβ-algebra A
gives rise to a tracial version of sequentially split β-homomorphism from
AβΞ±βG to Mβ£Gβ£β(A) and the tracial Rokhlin property of an
inclusion Cβ-algebras AβP with a conditional expectation E:AβP 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