A new approach to Hilbert \(C^*\)-modules
type: Item
instance of: scholarly article
title: A new approach to Hilbert \(C^*\)-modules
zbMATH Open document ID: 0874.46037
author: David P. Blecher
published in: Mathematische Annalen
publication date: 1997-08-14T00:00:00Z
review text: There is a canonical operator space structure on a \(C^*\)-module. Just as with Hilbert spaces, we can ignore the `inner product' on a \(C^*\)-module when it is convenient, and work only with the metric structure. The inner product is determined by the norm, and may be easily recovered. It is shown that many of the basic constructions with \(C^*\)-modules are familiar operator space constructions. Many apparently new results about \(C^*\)-modules are obtained, some of them unobtainable without the use of operator spaces. We give two new characterizations of \(C^*\)-modules. We also give an Eilenberg-Watts theorem, and a fundamental Morita theorem characterizing strong Morita equivalence of \(C^*\)-algebras in terms of isomorphism of \(C^*\)-module categories. Such theorems had previously been obtained by M. Rieffel for \(W^*\)-algebras. We show that the norm of an element \(u\) of the interior tensor product \(Y\otimes_\theta Z\) of two \(C^*\)-modules equals the Haagerup tensor norm, and also equals the completely bounded (but not the bounded) norm of \(u\) regarded as an operator \(\overline Y\to Z\). Here \(\overline Y\) is the conjugate \(C^*\)-module. These realizations permit one to study categorical \(C^*\)-module notions in a way which is much closer to the traditional `rings and modules' framework. For example, we obtain the basic HOM-tensor relations.
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 46H25
Mathematics Subject Classification ID: 46L89
Mathematics Subject Classification ID: 46M05
Mathematics Subject Classification ID: 46M15
Mathematics Subject Classification ID: 46C50
Mathematics Subject Classification ID: 47L05
Mathematics Subject Classification ID: 47L30
zbMATH DE Number: 1031531
zbMATH Keywords: completely bounded norm
zbMATH Keywords: operator space structure
zbMATH Keywords: \(C^*\)-module
zbMATH Keywords: Eilenberg-Watts theorem
zbMATH Keywords: fundamental Morita theorem
zbMATH Keywords: strong Morita equivalence of \(C^*\)-algebras
zbMATH Keywords: isomorphism of \(C^*\)-module categories
zbMATH Keywords: interior tensor product
zbMATH Keywords: Haagerup tensor norm
zbMATH Keywords: categorical \(C^*\)-module
zbMATH Keywords: HOM-tensor relations
MaRDI profile type: Publication
full work available at URL: s002080050033
OpenAlex ID: W2018958639
DOI: 10.1007/S002080050033
Recommended article: Q4426086
Recommended article: Q4369830
Recommended article: Hilbert \(C^*\)-modules: A useful tool
Recommended article: Q3967989
Recommended article: Q4796005
Recommended article: Hilbert \(C^*\)-modules and their bounded module maps
Recommended article: Q4328332
Recommended article: On full Hilbert \(C*\)-modules
Recommended article: Q3523867
Recommended article: A Hilbert \(C^*\)-module with extremal properties
description: scientific article; zbMATH DE number 1031531