The Radon-Nikodym theorem for non-commutative \(L^p\)-spaces
type: Item
instance of: scholarly article
title: The Radon-Nikodym theorem for non-commutative \(L^p\)-spaces
zbMATH Open document ID: 1196.46049
author: Hideaki Izumi
published in: Nihonkai Mathematical Journal
publication date: 2009-06-23T00:00:00Z
review text: In his doctoral dissertation [``Non-commutative \(L^p\)-spaces constructed by the complex interpolation method'' (Tohoku Mathematical Publications 9; Sendai:\ Tohoku Univ.) (1998; Zbl 0921.46070)], followed by a series of papers on the same subject, the author used complex interpolation to define families of noncommutative \(L^p\)-spaces over an arbitrary von Neumann algebra with a weight. These constructions generalized those of Kosaki, who defined them for a \(\sigma\)-finite von Neumann algebra with a state, and that of Terp, who dealt with a general algebra, but considered only the symmetric version. Since all these spaces are isometrically isomorphic to Haagerup spaces, they do not depend on the choice of the weight. Nevertheless, obtaining an explicit formula for the isometry between \(L^p_{(\alpha)}({\mathcal M},\varphi)\) and \(L^p_{(\alpha)}({\mathcal M},\psi)\) is nontrivial and is the subject of the present paper. Complicated calculations and a clever use of the balanced weight \(\phi \oplus \psi\) on the von Neumann algebra \(M_2({\mathcal M})\) lead to an isometry \(U^{\psi,\varphi}_{p,(\alpha)}\) that is natural in the sense of being the inverse of \(U^{\varphi,\psi}_{p,(\alpha)}\) and satisfying the chain rule \(U^{\theta,\psi}_{p,(\alpha)}U^{\psi,\varphi}_{p,(\alpha)}=U^{\theta,\varphi}_{p,(\alpha)}\).
reviewed by: Stanisław Goldstein
Mathematics Subject Classification ID: 46L52
Mathematics Subject Classification ID: 46L51
Mathematics Subject Classification ID: 47L20
zbMATH DE Number: 5568733
zbMATH Keywords: modular theory
zbMATH Keywords: noncommutative integration
zbMATH Keywords: Connes' Radon-Nikodým cocycle
zbMATH Keywords: complex interpolation
zbMATH Keywords: noncommutative \(L_p\)-spaces
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5568733