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 \(0<\lambda\), \(\lambda'\le 1\).