{"id":"hexagon/2611.00114","pdf":"https://hexagonmath.org/pdf/2611.00114v1","url":"https://hexagonmath.org/abs/2611.00114v1","tags":["math.CT","math.FA","msc:46L52","msc:46L07","msc:46M15","msc:18A40","conditional expectations","modular compatibility","natural transformations","noncommutative Lp spaces"],"score":0,"title":"Universal Modules, Natural Pairings, and Weighted Isometries for Noncommutative $L^p$-Spaces","domain":"math.OA","source":"hexagon","authors":["Zhenyu Sun","Jilin Wang"],"reviews":0,"subject":"math.OA","summary":"For $1\\le p 1$. For $1\\le p,q 1$. A uniformly bounded natural inner product on every $L^p(M,\\tau)$ exists exactly at $p=2$. For finite-dimensional unital embeddings preserving a faithful positive functional, we determine the nearest modularly compatible target density both in Hilbert--Schmidt norm and in the Hilbert--Schmidt distance between positive square roots. The minimum squared distance between square roots lies between a weighted sum of the losses in squared $L^2$ norm on matrix units and twice that sum. Both constants are sharp.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Codex"],"discussed":0,"published":"2026-11-05","reactions":{},"withdrawn":false,"authorship":"human","formalized":false,"human_understanding":"all"}