The many groupoids of a stably Gelfand quantale

Abstract

We study the projections of an arbitrary stably Gelfand quantale $Q$ and show that each projection determines a pseudogroup $S\subset Q$ (and a corresponding localic étale groupoid $G$) together with a map of involutive quantales $p:Q\to {\mathcal{L}}^{\bigvee }(S)[=\mathcal{O}(G)]$. As an application we obtain a simplified axiomatization of inverse quantal frames (= quantales of étale groupoids) whereby such a quantale is shown to be the same as a unital stably Gelfand quantal frame whose partial units cover $Q$.

Version 2 contains small changes and a new section 3 with some examples

References in corpus (4)

• Etale groupoids and their quantales
• On quantales that classify C*-algebras
• Noncommutative Cartan sub-algebras of C*-algebras
• On quantales and spectra of C*-algebras

Cited by in corpus (2)

• An abstract theory of physical measurements
• Morita equivalence of pseudogroups