On the extension of positive maps to Haagerup non-commutative $L^p$-spaces

Abstract

Let $M$ be a von Neumann algebra, let $\phi$ be a normal faithful state on $M$ and let $L^p(M,\phi )$ be the associated Haagerup non-commutative $L^p$-spaces, for $1\le p\le \infty$. Let $D\in L^1(M,\phi )$ be the density of $\phi$. Given a positive map $T:M\to M$ such that $\phi \circ T\le C_1\phi$ for some $C_1\ge 0$, we study the boundedness of the $L^p$-extension $T_{p,\theta }:D^{\frac{1-\theta }{p}}M{D}^{\frac{\theta }{p}}\to L^p(M,\phi )$ which maps $D^{\frac{1-\theta }{p}}x{D}^{\frac{\theta }{p}}$ to $D^{\frac{1-\theta }{p}}T(x)D^{\frac{\theta }{p}}$ for all $x\in M$. Haagerup-Junge-Xu showed that $T_{p,\frac{1}{2}}$ is always bounded and left open the question whether $T_{p,\theta }$ is bounded for $\theta \ne \frac{1}{2}$. We show that for any $1\le p<2$ and any $\theta \in [0,2^{-1}(1-\sqrt{p-1})]\cup [2^{-1}(1+\sqrt{p-1}),1]$, there exists a completely positive $T$ such that $T_{p,\theta }$ is unbounded. We also show that if $T$ is $2$-positive, then $T_{p,\theta }$ is bounded provided that $p\ge 2$ or $1\le p<2$ and $\theta \in [1-p/2,p/2]$.

Revised version, to appear in Forum of Mathematics, Sigma