Memoirs of the American Mathematical Society 2003; 68 pp; softcover Volume: 163 ISBN10: 0821832719 ISBN13: 9780821832714 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 noncommutative \(L^p\)spaces are obtained, such as the \(p\)BanachSaks 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 \(\mathcal N\) infinitedimensional, 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 \(0<\lambda\), \(\lambda'\le 1\). Readership Graduate students and research mathematicians interested in functional analysis and operator theory. Table of Contents  Introduction
 The modulus of uniform integrability and weak compactness in \(L^1(\mathcal N)\)
 Improvements to the main theorem
 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 semifinite
 \(L^p(\mathcal N)\)isomorphism results for \(\mathcal N\) a type III hyperfinite or a free group von Neumann algebra
 Bibliography
