Surjective separating maps on noncommutative $L^p$-spaces

Abstract

Let $1\le p<\infty$ and let $T:L^p(\mathcal{M})\to L^p(\mathcal{N})$ be a bounded map between noncommutative $L^p$-spaces. If $T$ is bijective and separating (i.e., for any $x,y\in L^p(\mathcal{M})$ such that $x^{\ast }y=x{y}^{\ast }=0$, we have $T(x{)}^{\ast }T(y)=T(x)T(y{)}^{\ast }=0$), we prove the existence of decompositions $\mathcal{M}={\mathcal{M}}_1\stackrel{\infty }{\oplus }{\mathcal{M}}_2$, $\mathcal{N}={\mathcal{N}}_1\stackrel{\infty }{\oplus }{\mathcal{N}}_2$ and maps $T_1:L^p({\mathcal{M}}_1)\to L^p({\mathcal{N}}_1)$, $T_2:L^p({\mathcal{M}}_2)\to L^p({\mathcal{N}}_2)$, such that $T=T_1+T_2$, $T_1$ has a direct Yeadon type factorisation and $T_2$ has an anti-direct Yeadon type factorisation. We further show that $T^{-1}$ is separating in this case. Next we prove that for any $1\le p<\infty$ (resp. any $1\le p\ne 2<\infty$), a surjective separating map $T:L^p(\mathcal{M})\to L^p(\mathcal{N})$ is $S^1$-bounded (resp. completely bounded) if and only if there exists a decomposition $\mathcal{M}={\mathcal{M}}_1\stackrel{\infty }{\oplus }{\mathcal{M}}_2$ such that $T{|}_{L^p({\mathcal{M}}_1)}$ has a direct Yeadon type factorisation and ${\mathcal{M}}_2$ is subhomogeneous.

Accepted for publication in Mathematische Nachrichten