Sequentially split $*$-homomorphisms between $\mathrm{C}^*$-algebras

TL;DR

This paper introduces sequentially split *-homomorphisms between C*-algebras, showing they preserve many approximation properties and arise naturally in group actions with the Rokhlin property, leading to new insights and generalizations.

Contribution

It defines and studies sequentially split *-homomorphisms, demonstrating their role in preserving approximation properties and their natural occurrence in Rokhlin actions, extending existing theory.

Findings

Approximation properties pass from target to domain algebra.

Sequentially split homomorphisms arise from Rokhlin actions.

New results include a generalized K-theory formula and duality results.

Abstract

We define and examine sequentially split $*$-homomorphisms between $\mathrm{C}^*$-algebras and $\mathrm{C}^*$-dynamical systems. For a $*$-homomorphism, the property of being sequentially split can be regarded as an approximate weakening of being a split-injective inclusion of $\mathrm{C}^*$-algebras. We show for a sequentially split $*$-homomorphism that a multitude of $\mathrm{C}^*$-algebraic approximation properties pass from the target algebra to the domain algebra, including virtually all important approximation properties currently used in the classification theory of $\mathrm{C}^*$-algebras. We also discuss various settings in which sequentially split $*$-homomorphisms arise naturally from context. One particular class of examples arises from compact group actions with the Rokhlin property. This allows us to recover and extend the presently known permanence properties of Rokhlin actions with a unified conceptual approach and a simple proof. Moreover, this perspective allows us to obtain new results about such actions, such as a generalization of Izumi's original $K$-theory formula for the fixed point algebra, or duality between the Rokhlin property and approximate representability.