On one side, a very recent paper of Chris Heunen and Andre Kornell "Axioms for the category of Hilbert spaces" (Arxiv:2109.7418v1) 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. Gérard Lang

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.