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Column rigidity in noncommutative $\L^1$-spaces and completely isometric embeddings of Hilbertian operator spaces

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

Contributed by Cédric ARHANCET ↗

Version 1 / Nov 03, 2026 / CC BY-SA 4.0

Abstract

We study exact embedding obstructions for Hilbertian operator spaces in noncommutative L1\mathrm{L}^1-spaces. We prove a quantitative version of the column inequality, whose equality case forces the absolute values of the entries to be proportional. We deduce that an operator space embedded completely isometrically in a noncommutative L1\mathrm{L}^1-space is a row Hilbert space whenever its first-column norms at the two-by-two level are Hilbert-Schmidt norms. This gives an exact embedding criterion for sums of weighted row and column spaces, including coordinate-dependent weights which are ordered in the same direction. In particular, Rn+Cn\mathrm{R}_n+\mathrm{C}_n does not embed completely isometrically into any noncommutative L1\mathrm{L}^1-space when n≥2n \geq 2. For OHn\mathrm{OH}_n, whose complete isometric embeddability is established in a companion paper, we retain the common-support and commutative-target obstructions. We also show that distortion one is approached by matrix models but is attained by none of them, and that the matrix size must tend to infinity as the distortion tends to one.

Provenance statement

The author acknowledges the use of AI tools for language polishing, exploratory mathematical discussions and feedback during the development and preparation of this manuscript.

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Version history

  1. v1Submitted by Cédric ARHANCETInitial depositCurrentNov 02, 2026