Banach embedding properties of non-commutative Lp-spaces

Let N and M be von Neumann algebras. It is proved that L-p(N) does not linearly topologically embed in L-p(M) for A infinite, M finite, 1 less than or equal to p < 2. The following considerably stronger result is obtained (which implies this, since the Schatten p-class C-p embeds in L-p(N) for M infinite). Theorem. Let 1 <= p < 2 and let X be a Banach space with a spanning set (X-ij) so that for some C greater than or equal to 1. (i) any row or column is C-equivalent to the usual l(2)-basis, (ii) (x(ik,jk)) is C-eqaivalent to the usual l(p)-basis, for any i(1) < i(2) < (...) and j(1) < j(2) < (. . .). Then X is not isomorphic to a subspace of L-p(M), for 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(M) containing l(p) isomorphically. The spaces L-p(N) are classified up to Banach isomorphism (i.e., linear homeomorphism), for N infinite-dimensional, hyperfinite and semifinite, 1 less than or equal to p < infinity, p not equal 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 less than or equal to p < infinity that L-p(M) is completely isomorphic to L-p(M) if N and M are the algebras associated to free groups, or if N and M are injective factors of type III lambda and III lambda' for 0 < lambda, lambda' less than or equal to 1.