Mixed $q$-deformed Araki–Woods von Neumann algebras

Abstract

Given a strongly continuous orthogonal representation $(U_t{)}_{t\in \mathbb{R}}$ of $\mathbb{R}$ on a real Hilbert space ${\mathcal{H}}_{\mathbb{R}}$, a decomposition ${\mathcal{H}}_{\mathbb{R}}:={⨁}_{i\in N}{\mathcal{H}}_{\mathbb{R}}^{(i)}$ consisting of invariant subspaces of $(U_t{)}_{t\in \mathbb{R}}$ and an appropriate matrix $((q_{ij}){)}_{N\times N}$ of real parameters, we associate representations of the mixed commutation relations on twisted Fock spaces. The associated von Neumann algebras are (usually) non-tracial and are generalizations of those constructed by Bożejko–Speicher and Hiai. We investigate the factoriality of these von Neumann algebras. Along the process, we show that the generating abelian subalgebras associated with the blocks of the aforesaid decomposition are strongly mixing masas when they admit appropriate conditional expectations. On the contrary, the generating abelian algebras which fail to admit appropriate conditional expectations are quasi-split. We also discuss non-injectivity and the Haagerup approximation property.