The Fredholm property for groupoids is a local property

TL;DR

This paper proves that the Fredholm property for groupoids is a local characteristic, allowing for easier verification in practical applications by examining local reductions.

Contribution

It extends the Fredholm property definition to locally compact groupoids and establishes its locality, enabling practical verification through local reductions.

Findings

Fredholm property is local for groupoids.

Reductions of groupoids determine the Fredholm property.

Primitive spectrum decomposes into local spectra.

Abstract

Fredholm Lie groupoids were introduced by Carvalho, Nistor and Qiao as a tool for the study of partial differential equations on open manifolds. This article extends the definition to the setting of locally compact groupoids and proves that \enquote{the Fredholm property is local}. Let $\mathcal{G} \rightrightarrows X$ be a topological groupoid and $(U_i)_{i\in I}$ be an open cover of $X$. We show that $\mathcal{G}$ is a Fredholm groupoid if, and only if, its reductions $\mathcal{G}^{U_i}_{U_i}$ are Fredholm groupoids for all $i \in I$. We exploit this criterion to show that many groupoids encountered in practical applications are Fredholm. As an important intermediate result, we use an induction argument to show that the primitive spectrum of $C^*(\mathcal{G})$ can be written as the union of the primitive spectra of all $C^*(\mathcal{G}|_{U_i})$, for $i \in I$.