Some results on continuous deformed free group factors

Abstract

We construct a Fock space associated to a symmetric function $Q:U\times U \to (-1,1)$, where $U$ is a nonempty open subset of $\mathbb R^j$ for some $j$. Namely, we will have operator-valued distributions $a(x)$ and $a^+(y)$ satisfying $a(x)a^+(y)-Q(x,y)a^+(y)a(x)=\delta(x-y)$. Analogous to the $q_{ij}$-Fock space of Bozejko and Speicher, we have field operators arising as the sum of the creation and annihilation operators. These operators generate a von Neumann algebra analogous to the free group factors, and we will show that they are factors which do not have property $\Gamma$. It was pointed out to us by an anonymous referee that this is a special case of a theorem of Krolak.