On certain operator modules and the normality of non-commutative varieties

Abstract

Within the framework of the theory of operator spaces and noncommutative geometry, we study problems well known in literature namely, we give a positive answer to the hyperinvariant sub- space problem for the operators intertwined with weighted bilateral shifts and other partials results in this context. In the category of Hilbert spaces we study the similarity problem of certain Hilbert modules by mean of the cohomology groups and we introduce a new class of operator (bi) modules wich generalizes the Hilbert C*-modules. Finaly, we study the noncom- mutative version of Serre-Swan theorem in the setting of Banach and Hilbert categories and we initiate the study of the classification of the Cuntz-Pimsner algebras by mean of thier associates C*-correspondences.