Embeddings of $\ell_p$ into non-commutative spaces

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 $L_{p,q}(\M, \tau)$, 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 $L_{p,q}(\M, \tau)$. As applications, we prove that for $0<p<\infty$ with $p \neq 2$ then $\ell_p$ cannot be strongly embedded into $L_p(\M,\tau)$. Thus providing a non-commutative extension of a result of Kalton for $0<p<1$ and a result of Rosenthal for $1\leq p <2$ on $L_p[0,1]$.