Groupoids and Hermitian Banach *-algebras

TL;DR

This paper investigates conditions under which twisted groupoid Banach *-algebras are Hermitian, establishing key properties and necessary conditions related to the structure of the groupoid and its fibers.

Contribution

It provides new criteria for the Hermitian property of twisted groupoid Banach *-algebras, including weak containment and fiber conditions, expanding understanding of their algebraic structure.

Findings

Hermitian groupoids satisfy the weak containment property

Hermitian property of $L^1(G,sigma)$ is linked to $L^1(G_sigma)$

Necessary conditions involve fibers $G^x_x$ for ample groupoids

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 $\mathcal{G}^x_x$.