Lack of isomorphic embeddings of ${\ell }_{p,q}$ into $L_{p,q}(\mathcal{M},\tau )$ over a noncommutative probability space

Abstract

We prove that the sequence space ${\ell }_{p,q}$ does not embed into $L_{p,q}(\mathcal{M},\tau )$ for any noncommutative probability space $(\mathcal{M},\tau )$, $1<p<\infty$, $1\le q<\infty$, $p\ne q$. Several applications to the isomorphic classification of noncommutative $L_{p,q}$-spaces are given, which extend and complement several earlier results.