Cohomology and extensions of Hilbert modules
type: Item
instance of: scholarly article
title: Cohomology and extensions of Hilbert modules
zbMATH Open document ID: 0833.46038
author: Douglas N. Clark
author: Jon F. Carlson
published in: Journal of Functional Analysis
publication date: 1995-07-31T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 46H25
Mathematics Subject Classification ID: 46M20
Mathematics Subject Classification ID: 47A45
zbMATH DE Number: 764009
zbMATH Keywords: Hilbert module
zbMATH Keywords: cohomology group
zbMATH Keywords: equivalence classes of short exact sequences
zbMATH Keywords: Ext
zbMATH Keywords: disk algebra
zbMATH Keywords: Hardy space
zbMATH Keywords: contractive Hilbert spaces
zbMATH Keywords: Sz.-Nagy and Foiaş models
reviewed by: H. Hogbe Nlend
MaRDI profile type: Publication
full work available at URL: jfan.1995.1032
OpenAlex ID: W2084941815
DOI: 10.1006/JFAN.1995.1032
description: scientific article; zbMATH DE number 764009