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Banach Embedding Properties of Non-Commutative $$L^p$$-Spaces
U. Haagerup, SDU Odense University, Denmark, H. P. Rosenthal, University of Texas, Austin, TX, and F. A. Sukochev, Flinders University of South Australia, Adelaide, Australia
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Memoirs of the American Mathematical Society
2003; 68 pp; softcover
Volume: 163
ISBN-10: 0-8218-3271-9
ISBN-13: 978-0-8218-3271-4
List Price: US$49 Individual Members: US$29.40
Institutional Members: US\$39.20
Order Code: MEMO/163/776

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

• The modulus of uniform integrability and weak compactness in $$L^1(\mathcal N)$$
• Complements on the Banach/operator space structure of $$L^p(\mathcal N)$$-spaces
• The Banach isomorphic classification of the spaces $$L^p(\mathcal N)$$ for $$\mathcal N$$ hyperfinite semi-finite
• $$L^p(\mathcal N)$$-isomorphism results for $$\mathcal N$$ a type III hyperfinite or a free group von Neumann algebra