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Banach Embedding Properties of Non-Commutative \(L^p\)-Spaces
U. Haagerup, SDU Odense University, Denmark, H. P. Rosenthal, University of Texas, Austin, TX, and F. A. Sukochev, Flinders University of South Australia, Adelaide, Australia

Memoirs of the American Mathematical Society
2003; 68 pp; softcover
Volume: 163
ISBN-10: 0-8218-3271-9
ISBN-13: 978-0-8218-3271-4
List Price: US$52
Individual Members: US$31.20
Institutional Members: US$41.60
Order Code: MEMO/163/776
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Let \(\mathcal N\) and \(\mathcal M\) be von Neumann algebras. It is proved that \(L^p(\mathcal N)\) does not linearly topologically embed in \(L^p(\mathcal M)\) for \(\mathcal N\) infinite, \(\mathcal M\) finite, \(1\le p<2\). The following considerably stronger result is obtained (which implies this, since the Schatten \(p\)-class \(C_p\) embeds in \(L^p(\mathcal N)\) for \(\mathcal N\) infinite).

Theorem. Let \(1\le p<2\) and let \(X\) be a Banach space with a spanning set \((x_{ij})\) so that for some \(C\ge 1\),

(i) any row or column is \(C\)-equivalent to the usual \(\ell^2\)-basis,

(ii) \((x_{i_k,j_k})\) is \(C\)-equivalent to the usual \(\ell^p\)-basis, for any \(i_1\le i_2 \le\cdots\) and \(j_1\le j_2\le \cdots\).

Then \(X\) is not isomorphic to a subspace of \(L^p(\mathcal M)\), for \(\mathcal M\) finite. Complements on the Banach space structure of non-commutative \(L^p\)-spaces are obtained, such as the \(p\)-Banach-Saks property and characterizations of subspaces of \(L^p(\mathcal M)\) containing \(\ell^p\) isomorphically. The spaces \(L^p(\mathcal N)\) are classified up to Banach isomorphism (i.e., linear homeomorphism), for \(\mathcal N\) infinite-dimensional, hyperfinite and semifinite, \(1\le p<\infty\), \(p\ne 2\). It is proved that there are exactly thirteen isomorphism types; the corresponding embedding properties are determined for \(p<2\) via an eight level Hasse diagram. It is also proved for all \(1\le p<\infty\) that \(L^p(\mathcal N)\) is completely isomorphic to \(L^p(\mathcal M)\) if \(\mathcal N\) and \(\mathcal M\) are the algebras associated to free groups, or if \(\mathcal N\) and \(\mathcal M\) are injective factors of type III\(_\lambda\) and III\(_{\lambda'}\) for \(0<\lambda\), \(\lambda'\le 1\).


Graduate students and research mathematicians interested in functional analysis and operator theory.

Table of Contents

  • Introduction
  • The modulus of uniform integrability and weak compactness in \(L^1(\mathcal N)\)
  • Improvements to the main theorem
  • Complements on the Banach/operator space structure of \(L^p(\mathcal N)\)-spaces
  • The Banach isomorphic classification of the spaces \(L^p(\mathcal N)\) for \(\mathcal N\) hyperfinite semi-finite
  • \(L^p(\mathcal N)\)-isomorphism results for \(\mathcal N\) a type III hyperfinite or a free group von Neumann algebra
  • Bibliography
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