On the classification of free Bogoljubov crossed product von Neumann algebras by the integers

TL;DR

This paper studies the structure and classification of free Bogoljubov crossed product von Neumann algebras generated by integer actions, providing new presentations, isomorphism criteria, and rigidity results.

Contribution

It introduces new descriptions of these algebras, establishes isomorphism and rigidity results, and characterizes strong solidity based on underlying representations.

Findings

Several presentations as amalgamated free products and cocycle crossed products.

Criteria for factoriality of the algebras.

Isomorphism and rigidity results for algebras from almost periodic representations.

Abstract

We consider crossed product von Neumann algebras arising from free Bogoljubov actions of the integers. We describe several presentations of them as amalgamated free products and cocycle crossed products and give a criterion for factoriality. A number of isomorphism results for free Bogoljubov crossed products are proved, focusing on those arising from almost periodic representations. We complement our isomorphism results by rigidity results yielding non-isomorphic free Bogoljubov crossed products and by a partial characterisation of strong solidity of a free Bogoljubov crossed products in terms of properties of the orthogonal representation from which it is constructed