Groupoids and Hermitian Banach *-algebras

Abstract

We study when the twisted groupoid Banach $\ast$-algebra $L^1(\mathcal{G},\sigma )$ is Hermitian. In particular, we prove that Hermitian groupoids satisfy the weak containment property. Furthermore, we find that for $L^1(\mathcal{G},\sigma )$ to be Hermitian it is sufficient that $L^1({\mathcal{G}}_{\sigma })$ is Hermitian. Moreover, if $\mathcal{G}$ is ample, we find necessary conditions for $L^1(\mathcal{G},\sigma )$ to be Hermitian in terms of the fibers ${\mathcal{G}}_x^x$.