Wick product for commutation relations connected with Yang-Baxter operators and new constructions of factors
The author analyzes a certain class of von Neumann algebras generated by self-adjoint elements \(\omega_i= a_i+ a^+_i\) where \(a_i a^+_j= \sum_{r,n} t^{ir}_{js} a^+_r a_s+ \delta_{ij}\mathrm{Id}\). Such algebras embed into some closures of the linear span of \(e_{i_1}\otimes e_{i_2}\otimes\cdots\otimes e_{i_n}\), where \(\{e_n\}\) is an orthogonal basis in a Hilbert space \({\mathcal H}\). The operator \(\psi(e_{i_1}\otimes e_{i_2}\otimes\cdots\otimes e_{i_n})\) representing the vector \(e_{i_1}\otimes e_{i_2}\otimes\cdots\otimes e_{i_n}\) is called the Wick product of operators \(\omega_{i_1}, \omega_{i_2},\dots, \omega_{i_n}\). An explicit description is given of the form of this product. An estimation of the operator norm of \(\psi(f)\) for \(f\in{\mathcal H}^n\) is obtained. These results are applied to prove that under the assumption \(\dim{\mathcal H}=\infty\) all the von Neumann algebras of the class are \(\mathrm{II}_1\) factors.
scientific article; zbMATH DE number 1442068