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

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.