Banach embedding properties of non-commutative L p -spaces. Memoirs of the American Mathematical Society; 776.

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

ISBN 10: 0821832719 / ISBN 13: 9780821832714

Edité par Providence, American Mathematical Society, 2003

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Ex-library in GOOD condition with library-signature and stamp(s). Some traces of use. Ehem. Bibliotheksexemplar mit Signatur und Stempel. GUTER Zustand, ein paar Gebrauchsspuren. C-00547 9780821832714 Sprache: Englisch Gewicht in Gramm: 350.

N° de réf. du vendeur 2483065

Détails bibliographiques

• Titre: Banach embedding properties of non-commutative L p -spaces. Memoirs of the American Mathematical Society; 776.
• Éditeur: Providence, American Mathematical Society
• Date de parution: 2003
• Langue: anglais
• ISBN à 10 chiffres: 0821832719
• ISBN à 13 chiffres: 9780821832714
• Reliure: Softcover
• Poids: 350 grammes
• Catalogues du vendeur:
  • SA MATHEMATIK

A propos de ce titre

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 $0<\lambda$, $\lambda'\le 1$.