Embeddings of ${\ell }_p$ into non-commutative spaces

Abstract

Let $\M$ be a semi-finite von Neumann algebra equipped with a faithful normal trace $\tau$. We study the subspace structures of non-commutative Lorentz spaces $L_{p,q}(\M, τ)$, extending results of Carothers and Dilworth to the non-commutative settings. In particular, we show that, under natural conditions on indices, ${\ell }_p$ can not be embedded into $L_{p,q}(\M, τ)$. As applications, we prove that for $0<p<\infty$ with $p\ne 2$ then ${\ell }_p$ cannot be strongly embedded into $L_p(\M,τ)$. Thus providing a non-commutative extension of a result of Kalton for $0<p<1$ and a result of Rosenthal for $1\le p<2$ on $L_p[0,1]$.

21 pages