Banach embedding properties of non-commutative Lp-spaces
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 t...
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Providence, R.I
American Mathematical Society
2003
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| Schriftenreihe: | Memoirs of the American Mathematical Society |
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| ISBN: | 0821832719, 9780821832714 |
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| Abstract | 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 $0<\lambda$, $\lambda'\le 1$. |
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| AbstractList | 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 $0<\lambda$, $\lambda'\le 1$. |
| Author | Sukochev, F. A. Haagerup, U. Rosenthal, Haskell P. |
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| BackLink | https://cir.nii.ac.jp/crid/1130000797291031808$$DView record in CiNii |
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| Notes | May 2003, volume 163, number 776 (third of 5 numbers) Includes bibliographical references (p. 67-68) |
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| Snippet | 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... |
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| SubjectTerms | Lp spaces Noncommutative function spaces Normed linear spaces Von Neumann algebras |
| Title | Banach embedding properties of non-commutative Lp-spaces |
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