Construction and uniqueness of the C*-Weyl algebra over a general pre-symplectic space

Abstract

A systematic approach to the C*-Weyl algebra W(E,σ) over a possibly degenerate pre-symplectic form σ on a real vector space E of possibly infinite dimension is elaborated in an almost self-contained manner. The construction is based on the theory of Kolmogorov decompositions for σ-positive-definite functions on involutive semigroups and their associated projective unitary group representations. The σ-positive-definite functions provide also the C*-norm of W(E,σ), the latter being shown to be *-isomorphic to the twisted group C*-algebra of the discrete vector group E. The connections to related constructions are indicated. The treatment of the fundamental symmetries is outlined for arbitrary pre-symplectic σ. The construction method is especially applied to the trivial symplectic form σ=0, leading to the commutative Weyl algebra over E, which is shown to be isomorphic to the C*-algebra of the almost periodic continuous function on the topological dual Eτ′ of E with respect to an arbitrary locally convex Hausdorff topology τ on E. It is demonstrated that the almost periodic compactification aEτ′ of Eτ′ is independent of the chosen locally convex τ on E, and that aEτ′ is continuously group isomorphic to the character group Ê of E. Applications of the results to the procedures of strict and continuous deformation quantizations are mentioned in the outlook.