C^∗-completions and the DFR-algebra

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.