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

Abstract

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

This is a finalized version. It will appear in J. Topology and Analysis. 16 pages

References in corpus (3)

• Completely positive maps of order zero
• Sequentially split $\ast$-homomorphisms between ${\mathrm{C}}^{\ast }$-algebras
• Rokhlin property and approximate representability for inclusions of index-finite type

Cited by in corpus (2)

• On dualities of actions and inclusions
• Rokhlin property and approximate representability for inclusions of index-finite type