Rigidity of pseudofunction algebras of ample groupoids

TL;DR

This paper demonstrates that Hausdorff, ample groupoids can be fully reconstructed from certain $L^p$-operator algebras, introducing new methods involving inverse semigroups and confirming a conjecture on Moore-Penrose inverses.

Contribution

It establishes the rigidity of pseudofunction algebras of ample groupoids and introduces a novel inverse semigroup construction for $L^p$-operator algebras.

Findings

Groupoids are recoverable from $I$-norm completions of $C_c( ext{groupoid})$.

New inverse semigroup construction from Moore-Penrose invertible partial isometries.

Confirmed conjecture on continuity of Moore-Penrose inverse in $L^p$-operator algebras.

Abstract

We show that a Hausdorff, ample groupoid $\mathcal{G}$ can be completely recovered from the $I$-norm completion of $C_c(\mathcal{G})$. More generally, we show that this is also the case for the algebra of symmetrized $p$-pseudofunctions, as well as for the reduced groupoid $L^p$-operator algebra, for $p\neq 2$. Our proofs are based on a new construction of an inverse semigroup built from Moore-Penrose invertible partial isometries in an $L^p$-operator algebra. Along the way, we verify a conjecture of Rako\v{c}evi\'{c} concerning the continuity of the Moore-Penrose inverse for $L^p$-operator algebras.