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

Palle E. T. Jorgensen·Daniil P. Proskurin·Yurii Samoilenko
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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\otimes h --> h \otimes 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., _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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Palle E. T. Jorgensen, Daniil P. Proskurin, Yurii Samoilenko. 2000-10-30. A family of *-algebras allowing Wick ordering: Fock representations and universal enveloping $C^*$-algebras. https://arxiv.org/abs/math/0010308

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