Topological full groups, invertible isometries, and automorphisms of groupoid algebras

TL;DR

This paper establishes a deep connection between the topological full group of a Hausdorff ample groupoid and invertible isometries in associated pseudofunction algebras, revealing new structural insights.

Contribution

It demonstrates that the topological full group coincides with homotopy classes of invertible isometries and characterizes automorphisms of pseudofunction algebras for effective groupoids.

Findings

Topological full group equals homotopy classes of invertible isometries.

Automorphisms form a split extension involving 1-cocycles.

Results apply to Hausdorff ample groupoids with compact unit space.

Abstract

We show that the topological full group of a Hausdorff ample groupoid with compact unit space coincides with the group of homotopy classes of invertible isometries in pseudofunction algebras associated with the groupoid. Moreover, if the groupoid $\mathcal{G}$ is also effective, then we show that the group of (inner) automorphisms in pseudofunction algebras is a split extension of the automorphisms (respectively, the topological full group) of $\mathcal{G}$ by the group of 1-cocycles (respectively, the 1-coboundaries).