Equivariant KK-theory and the continuous Rokhlin property

TL;DR

This paper introduces the continuous Rokhlin property for compact group actions on C*-algebras, characterizes it via asymptotic retracts, and explores its implications for KK-theory, crossed products, and classification of circle actions.

Contribution

It provides a KK-theoretical characterization of the continuous Rokhlin property, proves preservation of the UCT under such actions, and establishes a classification result for circle actions on Kirchberg algebras.

Findings

The continuous Rokhlin property is characterized by asymptotic retracts.

The UCT is preserved under crossed products and fixed point algebras for these actions.

KK$^{\mathbb{T}}$-equivalence classifies circle actions with the continuous Rokhlin property on Kirchberg algebras.

Abstract

We introduce and study the continuous Rokhlin property for actions of compact groups on C*-algebras. An important technical result is a characterization of the continuous Rokhlin property in terms of asymptotic retracts. As a consequence, we derive strong KK-theoretical obstructions to the continuous Rokhlin property. Using these, we show that the UCT is preserved under formation of crossed products and passage to fixed point algebras by such actions, even in the absence of nuclearity. Our analysis of the KK-theory of the crossed product allows us to prove a $\mathbb{T}$-equivariant version of Kirchberg-Phillips: two circle actions with the continuous Rokhlin property on Kirchberg algebras are conjugate whenever they are KK$^{\mathbb{T}}$-equivalent. In the presence of the UCT, this is equivalent to having isomorphic equivariant K-theory. We moreover characterize the KK$^{\mathbb{T}}$-theoretical invariants that arise in this way. Finally, we identify a KK$^{\mathbb{T}}$-theoretic obstruction to the continuous property, which is shown to be the only obstruction in the setting of Kirchberg algebras. We show by means of explicit examples that the Rokhlin property is strictly weaker than the continuous Rokhlin property.