\(C^{\ast}\)-completions and the DFR-algebra (Q2795599)

The Doplicher-Fredenhagen-Roberts algebra [\textit{S. Doplicher} et al., Commun. Math. Phys. 172, No.1, 187--220 (1995; Zbl 0847.53051)] is a model for non-commutative spacetime which describes uncertainty relations among spacetime coordinates dictated by gravitational effects at the Planck scale, where events cannot be localized with arbitrary precision due to the formation of event horizons.NEWLINENEWLINEThe present article provides a mathematical generalization of the DFR algebra to arbitrary spacetimes (Lorentzian manifolds, not necessarily flat Minkowski space) of arbitrary dimension. First, using Rieffel's deformation quantization theory, the authors define a pair of (unital and non-unital) \textit{\(C^*\)-algebras of canonical commutation relations} which is canonically associated to an arbitrary presymplectic space and which differs from other definitions existing in the literature. These algebras come along with two canonically defined (unique) \(C^*\)-norms, and, provided the symplectic form is non-degenerate, the non-unital one admits a unique non-degenerate irreducible \(C^*\)-algebra representation: the Schrödinger representation. As a second tool, they give a notion of \(C^*\)-completion for bundles of locally convex *-algebras with fiber seminorms (Fréchet *-algebra bundles) and study its universality and uniqueness properties.