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

TL;DR

The paper resolves, in the Haagerup non-commutative $L^p$-setting, the question of when a positive map $T:M\to M$ with $\boldsymbol{\rho}\circ T\le C_1\boldsymbol{\rho}$ extends to a bounded operator $T_{p,\theta}$ on $L^p(M,\boldsymbol{\rho})$ for general $\theta$. It shows a negative answer in the range $1\le p<2$ for $\theta$ away from $1/2$, by constructing finite-dimensional counterexamples and lifting them to infinite tensor products, and it provides positive extension results under 2-positivity for $p\ge 2$ (all $\theta$) and, for $1\le p\le 2$, when $\theta\in[1-p/2,p/2]$. The method combines a precise norm computation on semifinite and matrix models, Kadison–Schwarz arguments, and complex interpolation to propagate boundedness from $p=2$ (and $p=1$ in some cases) to a broader range of $p$ and $\theta$. The infinite tensor product technique then converts these finite-dimensional phenomena into global counterexamples, clarifying the limits of extension in the non-commutative $L^p$ landscape. Overall, the work delineates sharp parameter regimes for boundedness and demonstrates the sharpness via explicit countermodels.

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 non-commutative $L^p$-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 $T_{p,θ}\colon D^{\frac{1-θ}{p}} M D^{\fracθ{p}}\to L^p(M,\varphi)$ which maps $D^{\frac{1-θ}{p}} x D^{\fracθ{p}}$ to $D^{\frac{1-θ}{p}} T(x) D^{\fracθ{p}}$ for all $x\in M$. Haagerup-Junge-Xu showed that $T_{p,\frac12}$ is always bounded and left open the question whether $T_{p,θ}$ is bounded for $θ\not=\frac12$. We show that for any $1\leq p<2$ and any $θ\in [0,2^{-1}(1-\sqrt{p-1})]\cup[2^{-1}(1+\sqrt{p-1}), 1]$, there exists a completely positive $T$ such that $T_{p,θ}$ is unbounded. We also show that if $T$ is $2$-positive, then $T_{p,θ}$ is bounded provided that $p\geq 2$ or $1\leq p<2$ and $θ\in[1-p/2,p/2]$.

Theorems & Definitions (17)

• Lemma 2.1
• Lemma 3.1
• proof : Proof of Lemma \ref{['D-Gamma']}.
• Corollary 3.2
• Proposition 3.3
• Theorem 4.1
• proof
• Theorem 4.3
• proof
• Lemma 5.1