On Cohomology for Product Systems

TL;DR

This paper develops a cohomology theory for product systems of Hilbert bimodules using the Ext functor, explores explicit resolutions for certain algebraic dynamics, and demonstrates how 2-cocycles induce deformations of associated C*-algebras, including examples related to Q_N.

Contribution

It introduces a cohomology framework for product systems, explicitly constructs resolutions for algebraic dynamics, and shows how to deform C*-algebras via 2-cocycles.

Findings

Deformation of C*-algebras via 2-cocycles results in simple, purely infinite algebras.

Explicit resolutions are found for product systems related to algebraic dynamics.

Concrete examples of deformed Cuntz's algebra Q_N are analyzed.

Abstract

A cohomology for product systems of Hilbert bimodules is defined via the Ext functor. For the class of product systems corresponding to irreversible algebraic dynamics, relevant resolutions are found explicitly and it is shown how the underlying product system can be twisted by the 2-cocycles. In particular, this process gives rise to cohomological deformations of the C*-algebras associated with the product system. Concrete examples of deformations of the Cuntz's algebra Q_N arising this way are investigated and we show they are simple and purely infinite.