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

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., <a_i\phi,\psi>_T=<\phi,a_i^*\psi>_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.