C*-Completions and the DFR-Algebra

TL;DR

This paper develops a general framework for constructing C*-algebras, including the DFR-algebra for quantum spacetime, by extending C*-completion concepts to bundles and using deformation quantization.

Contribution

It introduces a novel method to build C*-algebras from Poisson bundles, encompassing the DFR-algebra as a special case.

Findings

Constructed a family of C*-algebras including the DFR-algebra

Extended C*-completion to bundles of algebras

Connected the construction to deformation quantization

Abstract

The aim of this paper is to present the construction of a general family of C*-algebras which includes, as a special case, the "quantum spacetime algebra" introduced by Doplicher, Fredenhagen and Roberts. It is based on an extension of the notion of C*-completion from algebras to bundles of algebras, compatible with the usual C*-completion of the appropriate algebras of sections, combined with a novel definition for the algebra of the canonical commutation relations using Rieffel's theory of strict deformation quantization. Taking the C*-algebra of continuous sections vanishing at infinity, we arrive at a functor associating a C*-algebra to any Poisson vector bundle and recover the original DFR-algebra as a particular example.