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Embeddings of ℓp into Non-commutative Spaces

Part of: Linear function spaces and their duals Selfadjoint operator algebras

Published online by Cambridge University Press:  09 April 2009

Narcisse Randrianantoanina
Narcisse Randrianantoanina
Affiliation:
Department of Mathematics and Statistics Miami UniversityOxford, Ohio 45056 USA e-mail: randrin@muohio.edu
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Abstract

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Let ℳ be a semi-finite von Neumann algebra equipped with a faithful normal trace τ. We prove a Kadec-Pelczyński type dichotomy principle for subspaces of symmetric space of measurable operators of Rademacher type 2. We study subspace structures of non-commutative Lorentz spaces Lp, q, (ℳ, τ), extending some results of Carothers and Dilworth to the non-commutative settings. In particular, we show that, under natural conditions on indices, ℓp cannot be embedded into Lp, q (ℳ, τ). As applications, we prove that for 0 < p < ∞ with p ≠ 2, ℓp cannot be strongly embedded into Lp(ℳ, τ). This provides a non-commutative extension of a result of Kalton for 0 < p < 1 and a result of Rosenthal for 1 ≦ p < 2 on Lp [0, 1].

Keywords

Lorentz spacesvon Neumann algebrasnon-commutative Lp-spaces.

MSC classification

Secondary: 46L52: Noncommutative function spaces 46L51: Noncommutative measure and integration 46E30: Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 74 , Issue 3 , June 2003 , pp. 331 - 350
Copyright
Copyright © Australian Mathematical Society 2003

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