Banach non-isomorphism of type III and semifinite noncommutative $L_p$-spaces

Abstract

Let $M$ be a nonzero type III von Neumann algebra with separable predual, and let $N$ be a semifinite von Neumann algebra. We prove that $M_*$ is not isomorphic to a subspace of $N_*$. For $1<p<\infty$, $p\ne2$, we prove that $L_p(M)$ is not isomorphic to a complemented subspace of $L_p(N)$. In particular, the Banach isomorphism class of $L_p(M)$ distinguishes type III algebras from semifinite algebras for $1\le p<\infty$, $p\ne2$.