Some results on continuous deformed free group factors

TL;DR

This paper constructs a new class of von Neumann algebras from a symmetric function-based Fock space, generalizing free group factors and demonstrating they are factors without property Gamma.

Contribution

It introduces a novel Fock space associated with a symmetric function, leading to new free-like von Neumann algebras with specific properties.

Findings

The constructed von Neumann algebras are factors.

They do not have property Gamma.

The construction generalizes 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.