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.

13 pages. This fixes some errors and updates the paper to reflect that the main results are already known

References in corpus (1)

• Non-commutative Lévy processes for generalized (particularly anyon) statistics

Cited by in corpus (1)

• Fock representations of $Q$-deformed commutation relations