Double Fell bundles over discrete double groupoids with folding

TL;DR

This paper introduces double Fell bundles and double C*-categories over discrete double groupoids with folding, providing algebraic tools for modeling noncommutative spaces, especially in finite dimensions.

Contribution

It constructs the concepts of double Fell bundles and double C*-categories, extending the algebraic framework for noncommutative geometry and generalizing the GNS construction.

Findings

Identified the algebra of sections with a double C*-algebra.

Established the dual category via Tomita-Takesaki involution.

Connected double Fell bundles to convolution algebras.

Abstract

In this paper we construct the notions of double Fell bundle and double C*-category for possible future use as tools to describe noncommutative spaces, in particular in finite dimensions. We identify the algebra of sections of a double Fell line bundle over a discrete double groupoid with folding with the convolution algebra of the latter. This turns out to be what one might call a double C*-algebra. We generalise the Gelfand-Naimark-Segal construction to double C*-categories and we form the dual category for a saturated double Fell bundle using the Tomita-Takesaki involution.