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

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We consider an abstract Wick ordering as a family of relations on elements aᵢ 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 Hₜ, <.,.>ₜ. If aᵢ* denotes the adjoint, i.e., <aᵢϕ,ψ>ₜ=<ϕ,aᵢ*ψ>ₜ, then we identify when <.,.>ₜ is positive semidefinite (the positivity question). In the case of deformations of the CCR-relations (the qij-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.