Factoriality of Bozejko-Speicher von Neumann algebras

We study the von Neumann algebra generated by q–deformed Gaussian elements l_i+l_i^* where operators l_i fulfill the q–deformed canonical commutation relations l_i l_j^-q l_j^ l_i=delta_{ij} for -1<q<1. We show that if the number of generators is finite, greater than some constant depending on q, it is a II_1 factor which does not have the property Gamma. Our technique can be used for proving factoriality of many examples of von Neumann algebras arising from some generalized Brownian motions, both for type II_1 and type III case.