On subspaces of non-commutative \(L_{p}\)-spaces.
type: Item
instance of: scholarly article
title: On subspaces of non-commutative \(L_{p}\)-spaces.
zbMATH Open document ID: 1056.46056
DOI: 10.1016/S0022-1236(03)00045-4
author: Yves Raynaud
author: Quanhua Xu
published in: Journal of Functional Analysis
publication date: 2003-10-14T00:00:00Z
review text: The main purpose of the paper under review is to investigate structural aspects of subspaces of non-commutative (Haagerup) \(L^p\)-spaces associated with general von Neumann algebras. One of the main results is on uniform containment of finite-dimensional Schatten classes \(S_p^n\) which follows from a more general theorem that goes as follows: Let \(0<p<\infty\), \(p\neq 2\), \(\mathcal{A}\) be a von Neumann algebra and \(X\) be a subspace of \(L^p(\mathcal{A})\). Let \((F_n)\) be a sequence of finite-dimensional (quasi-)normed spaces. If \(X\) contains \(K\)-uniformly \(\ell_p^n(F_j)\) for \(n,j\geq 1\), then \(X\) contains a subspace isomorphic to \(\left( \oplus_j F_j\right)_p\). For \(1\leq p<\infty\), a complemented version of the above stated result also holds. This result has a forerunner in the case of \(\mathcal{A}=B(\ell_2)\) and \(F_n=S_p^n\) in [\textit{J. Arazy} and \textit{J. Lindenstrauss}, Compos. Math. 30, 81--111 (1975; Zbl 0302.47034)]. The main tool is the concrete description of ultraproducts of non-commutative \(L^p\)-spaces developed by the first author in [J. Oper. Theory 48, No. 1, 41--68 (2002; Zbl 1029.46102)]. More precisely, the authors carefully study elements of the ultraproduct \(L^p(\mathcal{A})_{\mathcal{U}}\) which are disjoint from \(L^p(\mathcal{A})\). Their techniques allow them to deduce several results that in turn provide the following version of a non-commutative analogue of the classical Kadec-Pełczyński dichotomy (which was only known previously for the semi-finite case): Let \(\mathcal{A}\) be a von Neumann algebra, \(2<p<\infty\), and \(X\) be a closed subspace of \(L^p(\mathcal{A})\). Then either \(X\) is isomorphic to a Hilbert space and complemented in \(L^p(\mathcal{A})\) or \(X\) contains a subspace isomorphic to \(\ell_p\) and complemented in \(L^p(\mathcal{A})\). The authors also include a section that generalizes the results described above to the case of operator spaces.
reviewed by: Narcisse Randrianantoanina
Mathematics Subject Classification ID: 46L52
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 46L07
zbMATH DE Number: 1992840
zbMATH Keywords: noncommutative \(L_p\)-spaces
zbMATH Keywords: ultrapowers
zbMATH Keywords: Schatten classes
zbMATH Keywords: equiintegrability
MaRDI profile type: Publication
cites work: The Dual Space of an Operator Algebra
cites work: Q4057191
cites work: Q3346801
cites work: Q3329946
cites work: The optimality of James’s distortion theorems
cites work: Integral mappings and the principle of local reflexivity for noncommutative \(L^1\)-spaces
cites work: Q4249466
cites work: The Grothendieck-Pietsch and Dvoretzky-Rogers theorems for operator spaces
cites work: Q3344722
cites work: Banach embedding properties of non-commutative 𝐿^{𝑝}-spaces
cites work: An operator inequality
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: Uniformly non-square Banach spaces
cites work: On weakly compact operators on \(C^*\)-algebras
cites work: Embeddings of non-commutative \(L_p\)-spaces into non-commutative \(L_1\)-spaces, \(1<p<2\)
cites work: Doobs inequality for non-commutative martingales
cites work: On \(\mathcal O\mathcal L_\infty\) structures of nuclear \(C^\ast\)-algebras
cites work: Noncommutative Burkholder/Rosenthal inequalities
cites work: Bases, lacunary sequences and complemented subspaces in the spaces $L_{p}$
cites work: Q3667432
cites work: Applications of the complex interpolation method to a von Neumann algebra: non-commutative \(L^ p\)-spaces
cites work: Applications of Uniform Convexity of Noncommutative L p -Spaces
cites work: On the continuity of the map \(\phi\) \(\to | \phi |\) from the predual of a \(W^*\)-algebra
cites work: Espaces de Banach stables
cites work: Q4136031
cites work: Global versus local asymptotic theories of finite-dimensional normed spaces
cites work: Notes on non-commutative integration
cites work: A characterization of completely 1-complemented subspaces of noncommutative \(L_ 1\)-spaces.
cites work: Almost completely isometric embeddings between preduals of von Neumann algebras
cites work: Q4215796
cites work: Q4431278
cites work: Non-commutative martingale inequalities
cites work: Q4422024
cites work: Kadec-Pełczyński decomposition for Haagerup \(L^p\)-spaces
cites work: Non-commutative martingale transforms
cites work: Q4422014
cites work: On the structure of 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: Theory of operator algebras I.
cites work: Banach Lattices with the Subsequence Splitting Property
description: scientific article; zbMATH DE number 1992840