Semidiscrete Hilbert spaces
type: Item
instance of: scholarly article
title: Semidiscrete Hilbert spaces
zbMATH Open document ID: 0697.46026
DOI: 10.1007/BF02170059
author: Lothar M. Schmitt
published in: Acta Mathematica Hungarica
publication date: 1989-01-01T00:00:00Z
review text: Extract from the paper: ``In 1938 Sz. Nagy characterized \(L^ 2\)-spaces of commutative \(W^*\)-algebras in the following way: A Hilbert space H, which is ordered by a selfdual cone \(H^+\) having the Riesz interpolation property, is isomorphic to \(L^ 2(X,\mu)\) for some measure space (X,\(\mu)\). Grothendieck has shown that the identity on \(L^ 2(X,\mu)\) can be approximated in the strong operator topology by factorisation through \(C^ m\) with positive linear maps. As it can be seen from a result of \textit{G. Wittstock} [Lect. Notes Phys. 29, 67-84 (1974 Zbl 0329.46007); Theorem 3.1] Grothendieck's property imples the Riesz interpolation property for any Hilbert space with selfdual cone \(H^+\). Hence it characterizes as well the spaces \(L^ 2(X,\mu)\). In this paper an analogous theory is formulated for \(L^ 2\)-spaces of semidiscrete \(W^*\)-algebras.''
Mathematics Subject Classification ID: 46L10
Mathematics Subject Classification ID: 47L30
Mathematics Subject Classification ID: 46C99
Mathematics Subject Classification ID: 46A40
zbMATH DE Number: 4142787
zbMATH Keywords: semidiscrete H-space
zbMATH Keywords: factorisation of identity
zbMATH Keywords: \(L^ 2\)-spaces of commutative \(W^*\)-algebras
zbMATH Keywords: Riesz interpolation property
zbMATH Keywords: Grothendieck's property
reviewed by: Constantine G. Lascarides
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4142787