Three-Point Trace Positivity:Schatten Norm Compression and Interpolating Metrics
Abstract
Let (𝒩,ν) be a finite tracial von Neumann algebra and set 𝒦r(P,Q)=ν((P1/2QP1/2)r). We prove that [𝒦r(Pi,Pj)]i,j=13≥0, 1/2≤ r≤1. The range 1/2<r<1 is new even for matrices, and order three is sharp for 1/2≤ r<1. This single mechanism resolves two problems. First, if (ℛ,ω) is finite tracial and H=[Hij]i,j=13∈ M3(ℛ)+, then [ω(|Hij|p)]i,j=13≥0, 1≤ p≤2, settling the Lin--van den Driessche conjecture on Schatten norm compression. Second, for a semifinite tracial von Neumann algebra (ℳ,τ), define dt(A,B)2=(τ(A)+τ(B))/2− |Bt/4At/4|2/t2/t. We prove that dt is a complete metric on L1(ℳ,τ)+ for 1≤ t≤2, including the previously open range 1<t<2 between the Hellinger and Bures endpoints, and that dt is a metric on L1(ℳ,τ)+ Longleftrightarrow t∈[1,2] or ℳ is abelian. Its topology and Cauchy sequences agree with those of L1. The same classification holds for unital tracial C*-algebras and, with the known t=0 obstruction, resolves Problem~2 of Komálovics and Molnár.
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BibTeX
@misc{vuong2026threepoint,
title = {{Three-Point Trace Positivity:Schatten Norm Compression and Interpolating Metrics}},
author = {Trung Dung Vuong and Hiroyuki Osaka},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11021},
eprint = {2610.11021},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11021},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Three-Point Trace Positivity:Schatten Norm Compression and Interpolating Metrics
AU - Trung Dung Vuong
AU - Hiroyuki Osaka
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.11021
UR - https://arxiv.org/abs/2610.11021
AB - Let (𝒩,ν) be a finite tracial von Neumann algebra and set 𝒦_r(P,Q)=ν((P^1/2QP^1/2)^r). We prove that [𝒦_r(P_i,P_j)]_i,j=1^3≥0, 1/2≤ r≤1. The range 1/2<r<1 is new even for matrices, and order three is sharp for 1/2≤ r<1. This single mechanism resolves two problems. First, if (ℛ,ω) is finite tracial and H=[H_ij]_i,j=1^3∈ M_3(ℛ)_+, then [ω(|H_ij|^p)]_i,j=1^3≥0, 1≤ p≤2, settling the Lin--van den Driessche conjecture on Schatten norm compression. Second, for a semifinite tracial von Neumann algebra (ℳ,τ), define d_t(A,B)^2=(τ(A)+τ(B))/2− |B^t/4A^t/4|_2/t^2/t. We prove that d_t is a complete metric on L^1(ℳ,τ)_+ for 1≤ t≤2, including the previously open range 1<t<2 between the Hellinger and Bures endpoints, and that d_t is a metric on L^1(ℳ,τ)_+ Longleftrightarrow t∈[1,2] or ℳ is abelian. Its topology and Cauchy sequences agree with those of L^1. The same classification holds for unital tracial C^*-algebras and, with the known t=0 obstruction, resolves Problem~2 of Komálovics and Molnár.
ER -