Equivariant bundles and absorption

TL;DR

This paper characterizes when equivariant bundles absorb strongly self-absorbing actions using almost equivariant maps, extending known results to the equivariant setting and providing new technical tools for analysis.

Contribution

It introduces a new characterization of absorption for strongly self-absorbing actions via almost equivariant completely positive maps and extends stability results to equivariant $C_0(X)$-algebras.

Findings

Characterization of absorption using almost equivariant maps.

Extension of stability criteria to equivariant $C_0(X)$-algebras.

Dimension condition on $X$ can be removed if bundle is locally trivial.

Abstract

For a locally compact group $G$ and a strongly self-absorbing $G$-algebra $(\mathcal{D},\delta)$, we obtain a new characterization of absorption of a strongly self-absorbing action using almost equivariant completely positive maps into the underlying algebra. The main technical tool to obtain this characterization is the existence of almost equivariant lifts for equivariant completely positive maps, proved in recent work of the authors. This characterization is then used to show that an equivariant $C_0(X)$-algebra with $\mathrm{dim}_{\mathrm{cov}}(X)<\infty$ is $(\mathcal{D},\delta)$-stable if and only if all of its fibers are, extending a result of Hirshberg, R{\o}rdam and Winter to the equivariant setting. The condition on the dimension of $X$ is known to be necessary, and we show that it can be removed if, for example, the bundle is locally trivial.