\'Etale categories, restriction semigroups, and their operator algebras

TL;DR

This paper introduces new operator algebras associated with étale categories and restriction semigroups, establishing their properties and connections to existing $C^*$-algebras, thus answering a previously posed question.

Contribution

It defines full and reduced non-self-adjoint operator algebras for étale categories and restriction semigroups, and connects these to $C^*$-algebras via semicrossed products.

Findings

Operator algebras coincide with $C^*$-algebras in specific cases

Introduces semicrossed product algebra for étale actions

Answers a question posed by Kudryavtseva and Lawson

Abstract

We define the full and reduced non-self-adjoint operator algebras associated with \'etale categories and restriction semigroups, answering a question posed by Kudryavtseva and Lawson in \cite{lawson}. Moreover, we define the semicrossed product algebra of an \'etale action of a restriction semigroup on a $C^*$-algebra, which turns out to be the key point when connecting the operator algebra of a restriction semigroup with the operator algebra of its associated \'etale category. We also prove that in the particular cases of \'etale groupoids and inverse semigroups our operator algebras coincide with the $C^*$-algebras of the referred objects.