On Diagonal bimodules of \'{e}tale groupoid $C^*$-algebras

TL;DR

This paper characterizes when diagonal bimodules of étale groupoid C*-algebras are spectral, linking spectrality to invariance under dual group actions or coactions, with applications to ideals and subalgebras.

Contribution

It establishes necessary and sufficient conditions for spectrality of diagonal bimodules in étale groupoid C*-algebras, connecting spectrality to group actions and coactions.

Findings

Spectrality of diagonal bimodules is characterized by invariance under dual group actions or coactions.

The framework applies to transformation groupoids from group and semigroup actions.

Characterization of spectrality for ideals and subalgebras containing the diagonal.

Abstract

We study diagonal bimodules of \'{e}tale groupoid $C^*$-algebras over their canonical diagonal subalgebras, and establish necessary and sufficient conditions for such a bimodule to be spectral-that is, determined by its spectrum. For a class of $\Gamma$-graded \'{e}tale groupoids, we prove that the spectrality of diagonal bimodules is equivalent to their invariance under the action of the dual group $\widehat{\Gamma}$ in the abelian case, or under the coaction of $\Gamma$ in the nonabelian case, on the groupoid $C^*$-algebras, both of which are induced by the underlying cocycle. This framework covers transformation groupoids arising from homeomorphism actions of countable groups, as well as from local homeomorphism actions of Ore semigroups. As applications, we characterize the spectrality of closed two-sided ideals and subalgebras that contain the diagonal subalgebra of \'{e}tale groupoid $C^*$-algebras.