Topological full groups, invertible isometries, and automorphisms of groupoid algebras
Eusebio Gardella, Mathias Palmstrøm, Hannes Thiel
TL;DR
The paper shows that for a Hausdorff ample groupoid with compact unit space, the topological full group $\mathsf{F}(\mathcal{G})$ is encoded in the invertible isometries of natural pseudofunction algebras, with $\mathbf{U}(\mathfrak{A}(\mathcal{G})) \cong C(\mathcal{G}^{(0)},\mathbb{T}) \rtimes \mathsf{F}(\mathcal{G})$ and $\mathsf{F}(\mathcal{G}) \cong \mathbf{U}(\mathfrak{A}(\mathcal{G}))/\mathbf{U}_0(\mathfrak{A}(\mathcal{G}))$, uniformly across $p\in[1,\infty)\setminus\{2\}$. The work constructs two canonical maps, $\Gamma$ from 1-cocycles to automorphisms and $\Omega$ from automorphisms to groupoid automorphisms, establishing split-exact sequences that connect cocycle data and inner automorphisms with the full group of the groupoid. For effective groupoids with compact unit spaces, these yield induced split-exact sequences for inner and outer automorphisms, providing a robust framework that ties algebraic automorphisms of the pseudofunction algebras directly to the underlying groupoid symmetries. The results imply rigidity phenomena and lay groundwork for classifying groupoids via their associated algebras, with consequences for amenability of unitary groups and potential equivalences of groupoids from isomorphisms of the corresponding algebras.
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).