Sharp tangent inequalities in noncommutative $L_p$-spaces

Abstract

Let $\mathcal{M}$ be a von Neumann algebra equipped with a faithful normal semifinite weight, and let $L_p(\mathcal{M})$ denote the associated Haagerup noncommutative $L_p$-space. For $1<p<\infty$, let $q=p/(p-1)$. For $z\in L_q(\mathcal{M})$ and $y\in L_p(\mathcal{M})$, denote by $⟨z,y⟩=\mathrm{Re}\mathrm{Tr}(z^{\ast }y)$ the canonical real duality pairing between $L_q(\mathcal{M})$ and $L_p(\mathcal{M})$. Define the normalized duality map by $J_p(x)=\Vert x{\Vert }_p^{2-p}u|x{|}^{p-1}$ for $x=u|x|\ne 0$, and set $J_p(0)=0$. We establish the sharp tangent inequality $$\Vert x+y{\Vert }_p^2\ge \Vert x{\Vert }_p^2+2⟨J_p(x),y⟩+(p-1)\Vert y{\Vert }_p^2,x,y\in L_p(\mathcal{M}),1<p\le 2.$$ The reverse inequality holds for $2\le p<\infty$, and the constant $p-1$ is optimal. Our inequality extends Xu's tangent inequality \cite{Xu89} from classical $L_p$-spaces to general noncommutative $L_p$-spaces and provides the sharp tangent formulation of the Ricard--Xu convexity inequality \cite{RX16}. When specialized to Schatten $p$-classes, it also yields the corresponding tangent formulation of the Ball--Carlen--Lieb convexity inequality \cite{BCL94}. Under a suitable positivity assumption, we further establish a sharp complement to Xu's tangent inequality for Schatten $p$-classes in the range $0<p<1$.

19 pages. All comments are welcome!