Cohomology and extensions of Hilbert modules
A Hilbert module is defined as a separable Hilbert space which is a module over a function algebra, with a jointly continuous action. In the present paper, the authors introduce a familiar object from homological algebra, the cohomology group Ext, into the study of Hilbert modules. By definition, if \(K\) and \(H\) are Hilbert modules over a function algebra \(A\), \(\text{Ext} (K,H)\) is the group whose elements are equivalence classes of short exact sequences \[ 0\to H@> \alpha>> J@> \beta>> K\to 0, \] where \(J\) is a Hilbert module, \(\alpha\), \(\beta\) are Hilbert module maps. Standard homological algebra results are obtained as well as the description of Ext in particular cases involving the disk algebra, the Hardy space and contractive Hilbert spaces in connection with Sz.-Nagy and Foiaş models.
equivalence classes of short exact sequences
scientific article; zbMATH DE number 764009