A family of *-algebras allowing Wick ordering: Fock representations and universal enveloping C^*-algebras

2000/10/30 by Palle E. T. Jorgensen, Daniil P. Proskurin, Yurii Samoilenko
Mathematics · #math.QA #math.OA #msc:46L55 #msc:81S05

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published as Noncommutative Structures in Mathematics and Physics (Kiev, 2000) (S. Duplij and J. Wess, eds.), NATO Science Series II: Mathematics, Physics and Chemistry, vol. 22, Kluwer Academic Publishers, Dordrecht, 2001, pp. 321--329 · 9 pages, LaTeX2e document class "crckbked", NATO workshop proceedings (Kluwer) submission

arxiv created 2000/10/30 · arxiv updated 2009/11/30

Abstract

We consider an abstract Wick ordering as a family of relations on elements a_i and define *-algebras by these relations. The relations are given by a fixed operator T:h⊗ h --> h ⊗ h, where h is one-particle space, and they naturally define both a *-algebra and an inner-product space H_T, <.,.>_T. If a_i^* denotes the adjoint, i.e., <a_iϕ,ψ>_T=<ϕ,a_i^*ψ>_T, then we identify when <.,.>_T is positive semidefinite (the positivity question). In the case of deformations of the CCR-relations (the q_ij-CCR and the twisted CCR's), we work out the universal C*-algebras A, and we prove that, in these cases, the Fock representations of the A's are faithful.

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