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On one side, a very recent paper of Chris Heunen and Andre Kornell "Axioms for the category of Hilbert spaces" (Arxiv:2109.7418v1 latest Arxiv version) offers a characterization of the category of (real and complex) Hilbert spaces using only six purely categorical axioms. On another side a well-known theorem of Maria Soler gives a purely algebraic characterization concerning the infinite dimensional Hilbert spaces on the star division rings of the real or complex numbers or quaternions. I would like to know if anyone sees a way to build a relation between these two results.

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    $\begingroup$ Thanks for asking this question. I didn't know Sol`er’s Theorem before. Here is a link to a survey: arxiv.org/pdf/math/9504224.pdf. the new paper you mention (link: arxiv.org/pdf/2109.07418.pdf) uses this theorem in their proposition 5. This is how they get the concrete field $\mathbb{R}$ or $\mathbb{C}$ out of the rather abstract categorical setting. So in a concrete sense, there is a relation between these two results... $\endgroup$
    – Uri Bader
    Sep 17, 2021 at 12:30
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    $\begingroup$ I find this question very superficial, considering that Soler's result is the main ingredient in the proof of Heunen and Kornell, as a short look at the paper teaches. $\endgroup$
    – M Mueger
    Nov 4, 2021 at 15:46

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