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

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).