Embeddings of non-commutative \(L^p\)-spaces into preduals of finite von Neumann algebras
type: Item
instance of: scholarly article
title: Embeddings of non-commutative \(L^p\)-spaces into preduals of finite von Neumann algebras
zbMATH Open document ID: 1155.46032
author: Narcisse Randrianantoanina
published in: Israel Journal of Mathematics
publication date: 2008-05-27T00:00:00Z
review text: This paper studies the embedding problem of noncommutative \(L_p\)-spaces. The first main result states that, for any \(\sigma\)-finite von Neumann algebra \(R\), there exists a finite von Neumann algebra \(M\) such that \(L_p(R)\) isomorphically embeds into \(L_q(M)\) for any \(0<q<p<2\). The space \(L_p(R)\) used here is that defined by Haagerup. The author's proof heavily relies upon Haagerup's construction of \(L_p(R)\) as a space of measurable operators on a semifinite super-algebra. The second main result of the paper is the noncommutative analogue for semifinite algebras of Rosenthal's celebrated embedding theorem on reflexive subspaces of \(L_1\). Namely, if \(M\) is a semifinite von Neumann algebra and \(X\) a reflexive subspace of \(L_1(M)\), then \(X\) isomorphically embeds into \(L_p(M)\) for some \(1<p<2\). The key ingredient of the proof is an embedding theorem of \textit{G.\,Pisier} [Math.\ Ann.\ 276, 105--136 (1986; Zbl 0619.47016)]. Note that the noncommutative Rosenthal theorem in its full generality (for any von Neumann algebra) was recently proved by \textit{M.\,Junge} and \textit{J.\,Parcet} [Duke Math.\ J.\ 141, No.\,1, 75--122 (2008; Zbl 1176.46059)].
reviewed by: Quanhua Xu
Mathematics Subject Classification ID: 46L52
Mathematics Subject Classification ID: 46L51
Mathematics Subject Classification ID: 46L10
zbMATH DE Number: 5279150
zbMATH Keywords: noncommutative \(L_p\)-spaces
zbMATH Keywords: embedding
zbMATH Keywords: Rosenthal's theorem
MaRDI profile type: Publication
full work available at URL: s11856-008-0001-x
OpenAlex ID: W2330433136
cites work: Positive cones and \(L^p\)-spaces for von Neumann algebras
cites work: Q4039801
cites work: Non-commutative Banach function spaces
cites work: Noncommutative Kothe Duality
cites work: A uniform Kadec-Klee property for symmetric operator spaces
cites work: Q3766329
cites work: Generalized s-numbers of \(\tau\)-measurable operators
cites work: Subspaces of LC(H) and C p
cites work: Banach embedding properties of non-commutative 𝐿^{𝑝}-spaces
cites work: Les espaces \(L^ p\) d'une algebre de von Neumann definies par la derivee spatiale
cites work: Constructions of Non-Commutative Lp-Spaces with a Complex Parameter Arising from Modular Actions
cites work: On weakly compact operators on \(C^*\)-algebras
cites work: The norm of sums of independent noncommutative random variables in \(L_{p}(\ell_1)\)
cites work: Embeddings of non-commutative \(L_p\)-spaces into non-commutative \(L_1\)-spaces, \(1<p<2\)
cites work: Rosenthal's theorem for subspaces of noncommutative \(L_p\)
cites work: Q3667432
cites work: Q3736359
cites work: Applications of the complex interpolation method to a von Neumann algebra: non-commutative \(L^ p\)-spaces
cites work: Isomorphic characterizations of inner product spaces by orthogonal series with vector valued coefficients
cites work: Q4188284
cites work: Séries de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de Banach
cites work: Notes on non-commutative integration
cites work: Q4463849
cites work: Factorization of operators through \(L_{p\infty}\) or \(L_{p1}\) and non- commutative generalizations
cites work: Q4440813
cites work: Embeddings of ℓp into Non-commutative Spaces
cites work: Q4431852
cites work: On subspaces of non-commutative \(L_{p}\)-spaces.
cites work: On subspaces of L\(^p\)
cites work: Non-isomorphism of 𝐿_{𝑝}-spaces associated with finite and infinite von Neumann algebras
cites work: Embedding of non-commutative \(L^p\)-spaces: \(p<1\)
cites work: Q3876492
cites work: Q3995204
cites work: Embedding of \(C_q\) and \(R_q\) into noncommutative \(L_p\)-spaces, \(1 \leq p < q \leq 2\)
cites work: Analytic functions with values in lattices and symmetric spaces of measurable operators
DOI: 10.1007/S11856-008-0001-X
description: scientific article; zbMATH DE number 5279150