A short note on the continuous Rokhlin property and the universal coefficient theorem

TL;DR

This paper demonstrates that for actions of compact metrizable groups with the continuous Rokhlin property, the UCT in E-theory is preserved in crossed products and fixed point algebras, extending previous results to broader settings.

Contribution

It extends Gardella's result by showing UCT preservation under continuous Rokhlin actions for general compact groups, not just the circle, and relates E-theory classes of algebras.

Findings

UCT passes from A to A⋊_α G and A^α under the continuous Rokhlin property.

For circle actions, the E-theory class of A relates to that of A^α.

The result generalizes previous nuclear case to broader classes of C*-algebras.

Abstract

Let $G$ be a metrizable compact group, $A$ a separable C*-algebra and $\alpha$ a strongly continuous action of $G$ on $A$. Provided that $\alpha$ satisfies the continuous Rokhlin property, we show that the property of satisfying the UCT in E-theory passes from $A$ to the crossed product C*-algebra $A\rtimes_\alpha G$ and the fixed point algebra $A^\alpha$. This extends a result by Gardella in the case that $G$ is the circle and $A$ is nuclear. For circle actions on separable, unital C*-algebras with the continuous Rokhlin property, we establish a connection between the $E$-theory equivalence class of the coefficient algebra $A$ and the fixed point algebra $A^\alpha$.