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

Abstract Let M be a von Neumann algebra, let $\varphi $ be a normal faithful state on M and let $L^p(M,\varphi )$ be the associated Haagerup noncommutative $L^p$ -spaces, for $1\leq p\leq \infty $ . Let $D\in L^1(M,\varphi )$ be the density of $\varphi $ . Given a positive map $T\colon M\to M$ such that $\varphi \circ T\leq C_1\varphi $ for some $C_1\geq 0$ , we study the boundedness of the $L^p$ -extension $T_{p,\theta }\colon D^{\frac {1-\theta }{p}} MD^{\frac {\theta }{p}}\to L^p(M,\varphi )$ 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 $\theta \not ={\frac {1}{2}}$ . We show that for any $1\leq p<2$ 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 $p\geq 2$ or $1\leq p<2$ and $\theta \in [1-p/2,p/2]$ .