# Banach Embedding Properties of Non-Commutative \(L^{p}\)-Spaces

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*U. Haagerup; H. P. Rosenthal; F. A. Sukochev*

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 \(0<\lambda\), \(\lambda'\le
1\).

#### Table of Contents

# Table of Contents

## Banach Embedding Properties of Non-Commutative $L^{p}$-Spaces

- Contents v6 free
- Chapter 1. Introduction 18 free
- Chapter 2. The modulus of uniform integrability and weak compactness in L[sum(1)](N) 714 free
- Chapter 3. Improvements to the Main Theorem 2734
- Chapter 4. Complements on the Banach/operator space structure of L[sup(p)](N)-spaces 4148
- Chapter 5. The Banach isomorphic classification of the spaces L[sup(p)](N) for N hyperfinite semi-finite 4956
- Chapter 6. L[sup(P)](N)-isomorphism results for N a type III hyperfinite or a free group von Neumann algebra 6168
- Bibliography 6774