Operator-Valued Twisted Araki-Woods Algebras

TL;DR

This paper introduces operator-valued twisted Araki-Woods algebras, generalizing existing second quantization algebras, and develops a disintegration theory to analyze their structure and factoriality.

Contribution

It defines a new class of operator-valued algebras, extends disintegration techniques, and characterizes their modular theory and factoriality criteria.

Findings

Disintegration reduces algebra isomorphism to scalar case.

Characterization of modular theory for these algebras.

Criteria for factoriality of the constructed algebras.

Abstract

We introduce operator-valued twisted Araki-Woods algebras. These are operator-valued versions of a class of second quantization algebras that includes $q$-Gaussian and $q$-Araki-Woods algebras and also generalize Shlyakhtenko's von Neumann algebras generated by operator-valued semicircular variables. We develop a disintegration theory that reduces the isomorphism type of operator-valued twisted Araki-Woods algebras over type I factors to the scalar-valued case. Moreover, these algebras come with a natural weight, and we characterize its modular theory. We also give sufficient criteria that guarantee that factoriality of these algebras.