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

U. Haagerup SDU Odense University, Odense, Denmark
H. P. Rosenthal University of Texas, Austin, TX
F. A. Sukochev Flinders University of South Australia, Adelaide, Australia
Available Formats:
Electronic ISBN: 978-1-4704-0374-4
Product Code: MEMO/163/776.E
List Price: $52.00 MAA Member Price:$46.80
AMS Member Price: $31.20 Click above image for expanded view Banach Embedding Properties of Non-Commutative$L^p$-Spaces U. Haagerup SDU Odense University, Odense, Denmark H. P. Rosenthal University of Texas, Austin, TX F. A. Sukochev Flinders University of South Australia, Adelaide, Australia Available Formats:  Electronic ISBN: 978-1-4704-0374-4 Product Code: MEMO/163/776.E  List Price:$52.00 MAA Member Price: $46.80 AMS Member Price:$31.20
• Book Details

Memoirs of the American Mathematical Society
Volume: 1632003; 68 pp
MSC: 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 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.

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

Review Copy – for reviewers who would like to review an AMS book
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Volume: 1632003; 68 pp
MSC: 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 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.

• 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 semi-finite
• 6. $L^p(\mathcal {N})$-isomorphism results for $\mathcal {N}$ a type III hyperfinite or a free group von Neumann algebra
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
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