Reconstructing Etale Groupoids from Semigroups

Abstract

We reconstruct étale groupoids from semigroups of functions defined upon them, thus unifying reconstruction theorems such as [Kum86] & [Ren08] Kumjian-Renault's reconstruction from a groupoid C*-algebra.[Exe10] Exel's reconstruction from an ample inverse semigroup.[Ste19] Steinberg's reconstruction from a groupoid ring.[CGT19] Choi-Gardella-Thiel's reconstruction from a groupoid L p -algebra.Given a groupoid G, we denote the source and range of any g ∈ G byWe denote the units of G by G 0 and the composable pairs by G 2 , i.e.We denote the bisections of G byEquivalently, B ⊆ G is a bisection precisely when, for all g, h ∈ G,Arbitrary subsets of G are denoted by P(G) = {X : X ⊆ G}.

Representing Structured Semigroups on Etale Groupoid Bundles

“…Note that every normal subsemigroup Z is binormal, as ab ∈ Z then implies that aZb = abZ ⊆ ZZ ⊆ Z. However, the converse can fail, even with groupoid C*algebras -see [BC20,Example 7.3].…”

“…We will have more to say about U(S) in future work. For the moment we simply remark that ultrafilters are more natural to consider than arbitrary filters when dealing with groupoid C*-algebras and more general 'bumpy semigroups' -see [Bic19] and [BC20]. 10.1.…”

“…For example, if A is already known to be groupoid C*-algebra then this groupoid can be recovered just from the semigroup structure of S and D (as the groupoid of ultrafilters w.r.t. the 'domination relation' on S -see [BC20]). Even if A were a twisted groupoid C*-algebra or, equivalently, the C*-algebra of a Fell line bundle over a groupoid, then we could similarly recover the multiplicative part of the bundle just from S and D.…”

Dauns-Hofmann-Kumjian-Renault Duality for Fell Bundles and Structured C*-Algebras

“…Our original plan was to do the same work here with < * rather than <, [Bic21] being the first step in this direction. However, our collaboration with Clark in [BC20] revealed that < is a better replacement for < * . For one thing, < can be defined without any involution and thus makes sense in more general structures, like the Steinberg rings considered in [Bic20a].…”

“…On the other side, the Weyl groupoid also comes with a twist or, equivalently, a Fell line bundle. In recent years there has also been a big push to extend the Weyl groupoid construction in various directions -see [BCaH17], [BCW17], [CRST17], [CR18], [CGT19], [Ste19], [KM20], [BC20], [AdCC + 21] and [Bic21]. From this together with the vast literature on groupoid C*-algebras that has emerged over the past 30-40 years, it seems safe to say that étale groupoids are the best candidate for these long sought after 'non-commutative spaces'.…”