Three-Point Trace Positivity:Schatten Norm Compression and Interpolating Metrics

Abstract

Let $(\mathcal{N},\nu )$ be a finite tracial von Neumann algebra and set ${\mathcal{K}}_r(P,Q)=\nu ((P^{1/2}Q{P}^{1/2}{)}^r)$. We prove that $$[{\mathcal{K}}_r(P_i,P_j){]}_{i,j=1}^3\ge 0,1/2\le r\le 1.$$ The range $1/2<r<1$ is new even for matrices, and order three is sharp for $1/2\le r<1$. This single mechanism resolves two problems. First, if $(\mathcal{R},\omega )$ is finite tracial and $H=[H_{ij}{]}_{i,j=1}^3\in M_3(\mathcal{R}{)}_{+}$, then $$[\omega (|H_{ij}{|}^p){]}_{i,j=1}^3\ge 0,1\le p\le 2,$$ settling the Lin--van den Driessche conjecture on Schatten norm compression. Second, for a semifinite tracial von Neumann algebra $(\mathcal{M},\tau )$, define $$d_t(A,B{)}^2=\frac{\tau (A)+\tau (B)}{2}-\Vert B^{t/4}{A}^{t/4}{\Vert }_{2/t}^{2/t}.$$ We prove that $d_t$ is a complete metric on $L^1(\mathcal{M},\tau {)}_{+}$ for $1\le t\le 2$, including the previously open range $1<t<2$ between the Hellinger and Bures endpoints, and that $$d_t\text{ is a metric on }{L}^1(\mathcal{M},\tau {)}_{+}⟺t\in [1,2]\text{or}\mathcal{M}\text{ is abelian}.$$ Its topology and Cauchy sequences agree with those of $L^1$. The same classification holds for unital tracial $C^{\ast }$-algebras and, with the known $t=0$ obstruction, resolves Problem~2 of Komálovics and Molnár.

33 pages