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Étale categories, restriction semigroups, and their operator algebras

Natã Machado, Gilles G. de Castro

Abstract

We define the full and reduced non-self-adjoint operator algebras associated with étale categories and restriction semigroups, answering a question posed by Kudryavtseva and Lawson in \cite{lawson}. Moreover, we define the semicrossed product algebra of an étale 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 étale category. We also prove that in the particular cases of étale groupoids and inverse semigroups our operator algebras coincide with the $C^*$-algebras of the referred objects.

Étale categories, restriction semigroups, and their operator algebras

Abstract

We define the full and reduced non-self-adjoint operator algebras associated with étale categories and restriction semigroups, answering a question posed by Kudryavtseva and Lawson in \cite{lawson}. Moreover, we define the semicrossed product algebra of an étale action of a restriction semigroup on a -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 étale category. We also prove that in the particular cases of étale groupoids and inverse semigroups our operator algebras coincide with the -algebras of the referred objects.
Paper Structure (18 sections, 47 theorems, 171 equations)

This paper contains 18 sections, 47 theorems, 171 equations.

Table of Contents

  1. Étale categories and their operator algebras
  2. Operator algebras associated with étale categories
  3. The reduced operator algebra of an étale category
  4. Graph algebras
  5. Relational covering morphisms
  6. Restriction semigroups and their operator algebras
  7. Operator algebras associated with restriction semigroups
  8. The reduced operator algebra of a restriction semigroup
  9. The inverse semigroup case
  10. Restriction semigroup étale actions
  11. The canonical action of a restriction semigroup
  12. The invariance of the tight spectrum
  13. The semicrossed product algebra
  14. The category of germs of a restriction semigroup action
  15. The category of germs of the canonical action
  16. ...and 3 more sections

Key Result

Lemma 1.7

Let $X$ be a topological space and $Y \subseteq X$ a topological subspace of $X$. Suppose there exists a local homeomorphism $f: X\rightarrow Y$ such that $f(y)=y$ for every $y \in Y$. Then, $Y$ is open in $X$ and the extended map $f: X \rightarrow X$ is a local homeomorphism.

Theorems & Definitions (138)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Remark 1.4
  • Definition 1.5
  • Example 1.6: Transformation category
  • Lemma 1.7
  • proof
  • Proposition 1.8
  • proof
  • ...and 128 more