Published by American Mathematical Society, 2003

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Brand new. We distribute directly for the publisher. 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 $0<\lambda$, $\lambda'\le 1$. Bookseller Inventory #

**Synopsis:** 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 $0<\lambda$, $\lambda'\le 1$.

Title: **Banach Embedding Properties of ...**

Publisher: **American Mathematical Society**

Publication Date: **2003**

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ISBN 13: 9780821832714

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**Book Description **American Mathematical Society. Paperback. Condition: New. New copy - Usually dispatched within 2 working days. Seller Inventory # B9780821832714

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**Book Description **American Mathematical Society, United States, 2003. Paperback. Condition: New. Language: English . Brand New Book. 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. Seller Inventory # AAN9780821832714

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**Book Description **American Mathematical Society, 2003. PAP. Condition: New. New Book. Shipped from UK in 4 to 14 days. Established seller since 2000. Seller Inventory # CE-9780821832714

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**Book Description **American Mathematical Society, United States, 2003. Paperback. Condition: New. Language: English . Brand New Book. 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. Seller Inventory # AAN9780821832714

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ISBN 13: 9780821832714

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**Book Description **American Mathematical Society, 2003. Paperback. Condition: NEW. 9780821832714 This listing is a new book, a title currently in-print which we order directly and immediately from the publisher. For all enquiries, please contact Herb Tandree Philosophy Books directly - customer service is our primary goal. Seller Inventory # HTANDREE01498024

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**Book Description **Amer Mathematical Society, 2003. Paperback. Condition: Brand New. illustrated edition edition. 65 pages. 9.75x7.00x0.25 inches. In Stock. Seller Inventory # __0821832719

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**Book Description **Amer Mathematical Society, 2003. Mass Market Paperback. Condition: New. Seller Inventory # DADAX0821832719

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**Book Description **Amer Mathematical Society, 2003. Condition: Very Good. Ships from the UK. Great condition for a used book! Minimal wear. Seller Inventory # GRP97469889