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We study a pair of $C^*$-algebras by associating a $*$-homomorphism from $A$to $B$ allowing an approximate left-inverse to the sequence algebra of $A$ in amanner reminiscent of several tracial approximation properties. We areparticularly interested how regularity properties in the Elliott classificationprogram pass from $B$ to $A$. Among them, we show that the strict comparisonproperty and $\mathcal{Z}$-stability pass from $B$ to $A$ in our setting.

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

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