Locally $C^*$ Algebras, $C^*$ Bundles and Noncommutative Spaces

TL;DR

This paper generalizes Gelfand duality to noncommutative locally C*-algebras, representing them as sections of C*-bundles over topological spaces, and introduces the concept of noncommutative spaces inspired by physics applications.

Contribution

It extends Gelfand duality to locally C*-algebras and introduces noncommutative spaces via sheaves and bundles, with applications to quantum spacetime models.

Findings

Every locally C*- algebra is a section algebra of a C*-bundle.

Locally C*- algebras can define sheaves representing noncommutative spaces.

Construction of locally covariant quantum spacetimes from local nets of C*-algebras.

Abstract

This work provides a generalization of the Gelfand duality to the context of noncommutative locally $C^*$ algebras. Using a reformulation of a theorem proven by Dauns and Hofmann in the 60's we show that every locally $C^*$ algebra can be realized as the algebra of continuous sections of a $C^*$ bundle over a compactly generated topological space. This result is used then to show that on certain special cases locally $C^*$ algebras can be used to define certain sheaves of locally $C^*$ algebras that, inspired by the analogy with commutative geometry, we call noncommutative spaces. The last section provides some examples, motivated by mathematical physics, for this definition of noncommutative space. Namely we show that every local net of $C^*$ algebras defines a noncommutative space and, based on a loose generalization of the original construction by Doplicher, Fredenhagen and Roberts, construct what we propose to call a "locally covariant quantum spacetime".