Ground states on the ${\mathrm{C}}^{\ast }$-algebras of Fell bundles over étale groupoids

Let $p\text{\hspace{0.17em}⁣}:\mathcal{A}\to G$ be a Fell bundle over a locally compact Hausdorff second countable étale groupoid $G$ and let ${\mathrm{C}}^{\ast }(G;\mathcal{A})$ be the associated Fell bundle ${\mathrm{C}}^{\ast }$-algebra. Suppose ${\sigma }^c$ is the real dynamics on ${\mathrm{C}}^{\ast }(G;\mathcal{A})$ induced by a real-valued $1$-cocycle $c$. We establish an affine homeomorphism between the ${\sigma }^c$-ground state space of ${\mathrm{C}}^{\ast }(G;\mathcal{A})$ and the state space of ${\mathrm{C}}^{\ast }(G(Z);\mathcal{A}{\mathrm{\mid }}_{G(Z)})$, where $Z$ is the boundary set of $c$ and $G(Z)$ is the boundary groupoid. In particular, ${\mathrm{C}}^{\ast }(G;\mathcal{A})$ admits ${\sigma }^c$-ground states if and only if $Z\mathrm{\ne }\mathrm{\varnothing }$. We further investigate the relation between ground states and KMS${}_{\mathrm{\infty }}$ states and give a crystallization interpretation of the boundary ${\mathrm{C}}^{\ast }$-algebra ${\mathrm{C}}^{\ast }(G(Z);\mathcal{A}{\mathrm{\mid }}_{G(Z)})$ when the cocycle is locally constant. Finally, we apply our results to several classes of examples, including twisted ${\mathrm{C}}^{\ast }$-algebras of Deaconu–Renault groupoids and twisted higher-rank graph ${\mathrm{C}}^{\ast }$-algebras, obtaining explicit descriptions of their ground states.