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

TL;DR

This paper characterizes bijective separating maps between noncommutative L^p-spaces, showing their decompositions and factorization properties, and establishes conditions under which these maps are bounded or completely bounded.

Contribution

It provides a detailed structural analysis of surjective separating maps on noncommutative L^p-spaces, including their decompositions and boundedness criteria, extending prior understanding of such maps.

Findings

Existence of decompositions of von Neumann algebras related to separating maps.

Characterization of when separating maps are bounded or completely bounded.

Identification of conditions under which inverse maps are also separating.

Abstract

Let $1\leq p<\infty$ and let $T\colon L^p({\mathcal M})\to L^p({\mathcal N})$ 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 $x^*y=xy^*=0$, we have $T(x)^*T(y)=T(x)T(y)^*=0$), we prove the existence of decompositions ${\mathcal M}={\mathcal M}_1\mathop{\oplus}\limits^\infty{\mathcal M}_2$, ${\mathcal N}={\mathcal N}_1 \mathop{\oplus}\limits^\infty{\mathcal N}_2$ and maps $T_1\colon L^p({\mathcal M}_1)\to L^p({\mathcal N}_1)$, $T_2\colon L^p({\mathcal M}_2)\to L^p({\mathcal N}_2)$, 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 $1\leq p<\infty$ (resp. any $1\leq p\not=2<\infty$), a surjective separating map $T\colon L^p({\mathcal M})\to L^p({\mathcal N})$ is $S^1$-bounded (resp. completely bounded) if and only if there exists a decomposition ${\mathcal M}={\mathcal M}_1 \mathop{\oplus}\limits^\infty{\mathcal M}_2$ such that $T|_{L^p({\tiny {\mathcal M}_1})}$ has a direct Yeadon type factorisation and ${\mathcal M}_2$ is subhomogeneous.