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

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