2022/08/13 by Bauer, Wolfram, Fulsche, Robert
#46L60 #47L80 #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.2208.06591
The resolvent algebra R(X, σ) associated to a symplectic space (X, σ) was introduced by D. Buchholz and H. Grundling as a convenient model of the canonical commutation relation (CCR) in quantum mechanics. We first study a representation of R(ℂn, σ) with the standard symplectic form σ inside the full Toeplitz algebra over the Fock-Bargmann space. We prove that R(ℂn, σ) itself is a Toeplitz algebra. In the sense of R. Werner's correspondence theory we determine its corresponding shift-invariant and closed space of symbols. Finally, we discuss a representation of the resolvent algebra R(H, σ) for an infinite dimensional symplectic separable Hilbert space (H, σ). More precisely, we find a representation of R(H, σ) inside the full Toeplitz algebra over the Fock-Bargmann space in infinitely many variables.