HEXAGONbeta

Universal Modules, Natural Pairings, and Weighted Isometries for Noncommutative LpL^p-Spaces

Primarily human-written textHuman understanding: all partsmath.OA — Operator Algebrasmath.CT — Category Theorymath.FA — Functional Analysis

Contributed by Zhenyu Sun ↗, Jilin Wang

Version 1 / Nov 05, 2026 / CC BY 4.0

Abstract

For 1≤p<∞1\le p<\infty, we consider the functor LpL^p on finite tracial von Neumann algebras with trace-preserving embeddings, including nonunital ones. For a functor of unital contractive left modules with compatible distinguished vectors ξM\xi_M, finite direct sums of scalars determine the best constant in ∥xξM∥≤K∥x∥p\|x\xi_M\|\le K\|x\|_p. The pointed LpL^p functor is initial among the families satisfying this estimate with K=1K=1. Retaining only the bound on scalar points gives an atomic norm. On Cn\mathbb C^n with unit weights the norm of the identity from ℓnp\ell^p_n to this atomic space is asymptotic to p−1(log⁡n)1−1/pp^{-1}(\log n)^{1-1/p} for p>1p>1. For 1≤p,q<∞1\le p,q<\infty, every objectwise bounded natural bilinear family on Lp×LqL^p\times L^q has the form c τ(xy)+d τ(x)τ(y)c\,\tau(xy)+d\,\tau(x)\tau(y) for x,y∈Mx,y\in M, with c,dc,d independent of the object and c=0c=0 when 1/p+1/q>11/p+1/q>1. A uniformly bounded natural inner product on every Lp(M,τ)L^p(M,\tau) exists exactly at p=2p=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 L2L^2 norm on matrix units and twice that sum. Both constants are sharp.

Provenance statement

The manuscript was written by the human authors. OpenAI Codex was used for language editing, checks of notation and references, and preparation and compilation of the LaTeX submission source.

Tools used

OpenAI
Codex

References

  1. Uffe Haagerup, LpL^p -spaces associated with an arbitrary von N eumann algebra , Algèbres d'opérateurs et leurs applications en physique mathématique ( P roc. C olloq., M arseille, 1977), Colloq. Internat. CNRS, no. 274, CNRS, Paris, 1979, pp. 175--184.
  2. Uffe Haagerup, Marius Junge, and Quanhua Xu, A reduction method for noncommutative LpL_p -spaces and applications , Trans. Amer. Math. Soc. 362 (2010), no. 4, 2125--2165.
  3. Uffe Haagerup and Erling Størmer, Positive projections of von N eumann algebras onto JW -algebras , Rep. Math. Phys. 36 (1995), no. 2--3, 317--330.
  4. Hideaki Izumi, Natural bilinear forms, natural sesquilinear forms and the associated duality on non-commutative LpL^p -spaces , Internat. J. Math. 9 (1998), no. 8, 975--1039.
  5. Marius Junge, Zhong-Jin Ruan, and David Sherman, A classification for 2-isometries of noncommutative LpL_p -spaces , Israel J. Math. 150 (2005), 285--314.
  6. Marius Junge and David Sherman, Noncommutative LpL^p modules , J. Operator Theory 53 (2005), no. 1, 3--34.
  7. Hideki Kosaki, Applications of the complex interpolation method to a von N eumann algebra: non-commutative LpL^p -spaces , J. Funct. Anal. 56 (1984), no. 1, 29--78.
  8. Tom Leinster, A categorical derivation of L ebesgue integration , J. Lond. Math. Soc. (2) 107 (2023), no. 6, 1959--1982.
  9. Yu-Zhe Liu, Shengda Liu, Zhaoyong Huang, and Panyue Zhou, Normed modules and the categorification of integrations, series expansions, and differentiations , Sci. China Math. 69 (2026), no. 3, 571--614.
  10. G. G. Lorentz, On the theory of spaces Λ\Lambda , Pacific J. Math. 1 (1951), no. 3, 411--429.
Show all 18 referencesShow fewer references
  1. Edward Nelson, Notes on non-commutative integration , J. Funct. Anal. 15 (1974), no. 2, 103--116.
  2. Gilles Pisier, Martingale inequalities and operator space structures on LpL_p , Doc. Math. 19 (2014), 1367--1442.
  3. Gilles Pisier and Quanhua Xu, Non-commutative LpL^p -spaces , Handbook of the Geometry of Banach Spaces (William B. Johnson and Joram Lindenstrauss, eds.), vol. 2, North-Holland, Amsterdam, 2003, pp. 1459--1517.
  4. I. E. Segal, A non-commutative extension of abstract integration , Ann. of Math. (2) 57 (1953), no. 3, 401--457, Correction: Ann. of Math. (2) 58 (1953), 595--596.
  5. David Sherman, On the structure of isometries between noncommutative LpL^p spaces , Publ. Res. Inst. Math. Sci. 42 (2006), no. 1, 45--82.
  6. Masamichi Takesaki, Conditional expectations in von N eumann algebras , J. Funct. Anal. 9 (1972), no. 3, 306--321.
  7. Marianne Terp, Interpolation spaces between a von N eumann algebra and its predual , J. Operator Theory 8 (1982), no. 2, 327--360.
  8. Jun Tomiyama, On the projection of norm one in W∗W^\ast -algebras , Proc. Japan Acad. 33 (1957), no. 10, 608--612.

Version history

  1. v1 · 23 pagesSubmitted by Zhenyu SunInitial depositCurrentNov 05, 2026