Factoriality of von Neumann algebras connected with general commutation relations—finite dimensional case
Volume 73 / 2006
Abstract
We study a certain class of von Neumann algebras generated by selfadjoint elements $\omega _i=a_i+a_i^+$, where $a_i, a_i^+$ satisfy the general commutation relations: $$ a_ia_j^+ =\sum\limits_{\scriptstyle r,s} t{}^ i_j{}^r_s\, \, a_r^+a_s +\delta_{ij}Id. $$ We assume that the operator $T$ for which the constants $t{}^ i_j{}^r_s$ are matrix coefficients satisfies the braid relation. Such algebras were investigated in [BSp] and [K] where the positivity of the Fock representation and factoriality in the case of infinite dimensional underlying space were shown. In this paper we prove that under certain conditions on the number of generators our algebra is a factor. The result was obtained for $q$-commutation relations by P. Śniady [Snia] and recently by E. Ricard [R]. The latter proved factoriality without restriction on the dimension, but it cannot be easily generalized to the general commutation relation case. We generalize the result of Śniady and present a simpler proof. Our estimate for the number of generators in case $q>0$ is better than in [Snia].
