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, $1le p<2$. 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 $1le p<2$ and let $X$ be a Banach space with a spanning set $(x_{ij})$ so that for some $Cge 1$, (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 $i_1le i_2 lecdots$ and $j_1le j_2le cdots$. 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, $1le p
