On cohomology for product systems

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 2-cocycles. In particular, this process gives rise to cohomological deformations of the $C^{\ast }$-algebras associated with the product system. Concrete examples of deformations of the Cuntz’s algebra ${\mathcal{Q}}_{\mathbb{N}}$ arising this way are investigated, and we show that they are simple and purely infinite.