Electronic ISBN:  9781470403744 
Product Code:  MEMO/163/776.E 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 163; 2003; 68 ppMSC: Primary 46; 47;
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 noncommutative \(L^p\)spaces are obtained, such as the \(p\)BanachSaks 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 \(\mathcal N\) infinitedimensional, 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 \(0<\lambda\), \(\lambda'\le 1\).ReadershipGraduate students and research mathematicians interested in functional analysis and operator theory.

Table of Contents

Chapters

1. Introduction

2. The modulus of uniform integrability and weak compactness in $L^1(\mathcal {N})$

3. Improvements to the main theorem

4. Complements on the Banach/operator space structure of $L^p(\mathcal {N})$spaces

5. The Banach isomorphic classification of the spaces $L^p(\mathcal {N})$ for $\mathcal {N}$ hyperfinite semifinite

6. $L^p(\mathcal {N})$isomorphism results for $\mathcal {N}$ a type III hyperfinite or a free group von Neumann algebra


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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 noncommutative \(L^p\)spaces are obtained, such as the \(p\)BanachSaks 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 \(\mathcal N\) infinitedimensional, 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.

Chapters

1. Introduction

2. The modulus of uniform integrability and weak compactness in $L^1(\mathcal {N})$

3. Improvements to the main theorem

4. Complements on the Banach/operator space structure of $L^p(\mathcal {N})$spaces

5. The Banach isomorphic classification of the spaces $L^p(\mathcal {N})$ for $\mathcal {N}$ hyperfinite semifinite

6. $L^p(\mathcal {N})$isomorphism results for $\mathcal {N}$ a type III hyperfinite or a free group von Neumann algebra