On Diagonal bimodules of étale groupoid $C^*$-algebras

TL;DR

The paper develops a framework for diagonal bimodules over the diagonal of reduced étale groupoid $C^*$-algebras, introducing a spectrum-based notion of spectral bimodules and establishing when diagonal bimodules are determined by their spectrum. By leveraging the Fourier coefficients approximation property and cocycle-induced (co)actions, it proves that for a $\Gamma$-graded étale groupoid, spectrality of a diagonal bimodule is equivalent to invariance under either the dual action $\widehat{\Gamma}$ (abelian $\Gamma$) or the cocycle coaction (nonabelian $\Gamma$), with precise statements in terms of intersections with spectral subspaces $C_r^*(\mathcal{G})_t$ and projections $\Phi_t$. The results yield a robust spectral theorem for amenable principal groupoids, show that diagonal bimodules are generated by elements with bisection support, and characterize spectral closed two-sided ideals as dynamical ideals, establishing bijections between open wide subgroupoids and Cartan-type subalgebras. The framework encompasses transformation groupoids from group actions and Ore semigroups, providing concrete criteria for spectrality and dynamical ideal structure across a broad class of étale groupoid $C^*$-algebras.

Abstract

We study diagonal bimodules of étale 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 $Γ$-graded étale groupoids, we prove that the spectrality of diagonal bimodules is equivalent to their invariance under the action of the dual group $\widehatΓ$ in the abelian case, or under the coaction of $Γ$ 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 étale groupoid $C^*$-algebras.

Theorems & Definitions (61)

• Remark 3.1
• Definition 3.2
• Definition 3.3
• Proposition 3.4
• proof
• Theorem 3.5
• Example 3.6
• Example 3.7
• Definition 3.8
• Proposition 3.9