  Let N and M be von Neumann algebras. It is proved that L^p(N) does not Banach
embed in L^p(M) for N infinite, M finite, 1 < or = 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 N infinite). Theorem: Let 1 < or = p
< 2 and let X be a Banach space with a spanning set (x_{ij}) so that for some C
< or = 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 < 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 ell^p isomorphically. The spaces L^p(N) are
classified up to Banach isomorphism, for N infinite-dimensional, hyperfinite
and semifinite, 1 < or = p< infty, p not= 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 < or = p < infty that L^p(N) 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' < or = 1.
