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.

A C*-Algebra for Quantized Principal U(1)-Connections on Globally Hyperbolic Lorentzian Manifolds

“…For the case of symplectic vector spaces the theory of CCR-representations is well understood and details can be found in [BGP07,BG12,BR96]. The generalization to presymplectic vector spaces has been studied in [BHR04] …”

Field-Theoretic Weyl Deformation Quantization of Enlarged Poisson Algebras

“…(For twisted group algebras we defer the reader to the citations in Subsection 3.2, and for the Weyl algebra with degenerate σ see [30,31], and references therein.) The C * -norm on W(E, σ) is denoted by · .…”

“…In virtue of the linear independence of the W (f ), f ∈ E, β is a well-defined * -isomorphism from ∆(E, σ) onto ∆(E, σ). Extend norm-continuously using a result in [31].…”

Some Continuous Field Quantizations, Equivalent to the C*-Weyl Quantization

“…Here T is an element of the group symp(E, σ) of all symplectic transformations on the pre-symplectic space (E, σ). (Recall that T ∈ symp(E, σ) is an R-linear bijection on E with σ(f, g) = σ(T f, T g); [24]). With Eq.…”

“…may be realized in terms of a bivector field, applied to differentials. For this we introduce an arbitrary locally convex Hausdorff vector space topology τ on E. On the topological dual space E τ of E we consider the σ(E τ , E)-topology, and so the bidual is given by (E τ ) = E. According to [24] the commutative C*-Weyl algebra W(E, 0) is *-isomorphic to the C*-algebra of the almost periodic, σ(E τ , E)-continuous functions on E τ , and we may regard each element A ∈ W(E, 0) as an almost periodic function A : E τ → C. The Weyl elements W 0 (f ) are realized in terms of the periodic functions (3.11)…”