Wick product for commutation relations connected with Yang-Baxter operators and new constructions of factors

We analyze a certain class of von Neumann algebras generated by selfadjoint elements ωi = ai + a+i, for ai , a+i satisfying the general commutation relations: (formula presented) Such algebras can be continuously embedded into some closure of the set of finite linear combinations of vectors eil ⊗ . . . ⊗ eik, where {ei} is an orthonormal basis of a Hilbert space H. The operator which represents the vector eil ⊗ . . . ⊗ ein is denoted by ψ (eil ⊗ . . . ⊗ ein) and called the "Wick product" of the operators ω1l, . . . , ωin. We describe explicitly the form of this product. Also, we estimate the operator norm of ψ(f) for f ∈ H⊗n. Finally we apply these two results and prove that under the assumption dim H = ∞ all the von Neumann algebras considered are I I1 factors.