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The operator Hilbert space OH\mathrm{OH} embeds completely isometrically into a noncommutative L1\mathrm{L}^1-space

Primarily human-written textHuman understanding: all partsmath.FA — Functional Analysismath.OA — Operator Algebras

Contributed by Cédric ARHANCET ↗

Version 1 / Oct 26, 2026 / CC BY-SA 4.0

Abstract

We prove that Pisier's operator Hilbert space OH\mathrm{OH} embeds completely isometrically into the predual of a QWEP\mathrm{QWEP} von Neumann algebra, answering a longstanding question introduced by Pisier and later by Haagerup and Musat. The embedding is obtained as an ultraproduct of free probabilistic models hk=Λ12ckΛ12h_k = \Lambda^{\frac{1}{2}}c_k\Lambda^{\frac{1}{2}} realized in the free group factor VN(F∞)\mathrm{VN}(\mathbb{F}_\infty), with the trace and circular family scaled according to the mass of the truncated measure t−53dtt^{-\frac{5}{3}} \mathrm{d} t.

Provenance statement

This work grew out of a quest lasting several months, during which the author first investigated finite-dimensional obstructions before seeking a completely isometric embedding of $\mathrm{OH}$ into a noncommutative $\L^1$-space. AI tools supported exploratory mathematical discussions, the development and scrutiny of tentative arguments, and revision of parts of the manuscript. They also assisted with programming for numerical experiments involving nonlinear matrix equations, random matrix models and truncation asymptotics. These experiments informed the search for the construction and the comparison of the two norm computations. AI tools were also used for language polishing. The author takes full responsibility for the mathematical content and final presentation.

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Version history

  1. v1Submitted by Cédric ARHANCETInitial depositCurrentOct 26, 2026