Groupoids and Hermitian Banach *-algebras

Abstract

We study when the twisted groupoid Banach ∗-algebra L 1 (G, σ) is Hermitian. In particular, we prove that Hermitian groupoids satisfy the weak containment property. Furthermore, we find that for L 1 (G, σ) to be Hermitian it is sufficient that L 1 (Gσ) is Hermitian. Moreover, if G is ample, we find necessary conditions for L 1 (G, σ) to be Hermitian in terms of the fibers G x x .