Simplicity of inverse semigroup and étale groupoid algebras

In this paper, we prove that the algebra of an étale groupoid with totally disconnected unit space has a simple over field if and only is minimal effective function vanishes on every open subset null function. Previous work subject required to be also topologically principal in non-Hausdorff case, but do not. Furthermore, provide first examples not groupoids spaces. Our come from self-similar group actions uncountable groups. More generally, show essential (the quotient by ideal functions vanishing set), free, generalizing algebraic setting recent result for C⁎-algebras. The main application our description contracted inverse semigroup algebras, thereby answering question Munn seventies. Using Galois descent, simplicity algebras depends characteristic can lifted positive 0. We semigroups outside prescribed set prime characteristics.