Banach Embedding Properties of Non-Commutative Lp-Spaces

Let $mathcal N$ and $mathcal M$ be von Neumann algebras. It is proved that $Lp(mathcal N)$ does not linearly topologically embed in $Lp(mathcal M)$ 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 $Cp$ embeds in $Lp(mathcal N)$ for $mathcal N$ infinite). Theorem. Let $1le p<2$ and let $X$ be a Banach space with a spanning set $(xij)$ so that for some $Cge 1$, (i) any row or column is $C$-equivalent to the usual $ell2$-basis, (ii) $(xik,jk)$ is $C$-equivalent to the usual $ellp$-basis, for any $i1le i2 lecdots$ and $j1le j2le cdots$. Then $X$ is not isomorphic to a subspace of $Lp(mathcal M)$, for $mathcal M$ finite.Complements on the Banach space structure of non-commutative $Lp$-spaces are obtained, such as the $p$-Banach-Saks property and characterizations of subspaces of $Lp(mathcal M)$ containing $ellp$ isomorphically. The spaces $Lp(mathcal N)$ are classified up to Banach isomorphism (i.e., linear homeomorphism), for $mathcal N$ infinite-dimensional, hyperfinite and semifinite, $1le p<infty$, $pne 2$. 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 $1le p<infty$ that $Lp(mathcal N)$ is completely isomorphic to $Lp(mathcal M)$ 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 $0<lambda$, $lambda'le 1$.