Groupoids and Hermitian Banach *-algebras

Abstract

We study when the twisted groupoid Banach $*$-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 $\mathcal{G}^x_x$.