# An order theoretic characterization of spin factors

Lemmens, Bas, Roelands, Mark, Van Imhoff, Hent (2017) An order theoretic characterization of spin factors. The Quarterly Journal of Mathematics, 68 (3). pp. 1001-1017. ISSN 0033-5606. (doi:10.1093/qmath/hax010) (KAR id:57539)

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## Abstract

The famous Koecher–Vinberg theorem characterizes the Euclidean Jordan algebras among the finite dimensional order unit spaces as the ones that have a symmetric cone. Recently, Walsh gave an alternative characterization of the Euclidean Jordan algebras. He showed that the Euclidean Jordan algebras correspond to the finite dimensional order unit spaces (V, C, u) for which there exists a bijective map $$g : C° → C°$$ with the property that $$g$$ is antihomogeneous, that is, $$g (\lambda x) = \lambda^{-1}g(x)$$ for all $$\lambda > 0$$ and $$x \in C°$$, and $$g$$ is an order-antimorphism, that is, $$x ≤ c\space y$$ if and only if $$g(y) ≤ g (x)$$. In this paper, we make a first step towards extending this order theoretic characterization to infinite dimensional JB-algebras. We show that if $$(V, C, u)$$ is a complete order unit space with a strictly convex cone and $$dim V ≥ 3$$, then there exists a bijective antihomogeneous order-antimorphism $$g : C° → C°$$ if and only if $$(V, C, u)$$ is a spin factor.

Item Type: Article 10.1093/qmath/hax010 Q ScienceQ Science > QA Mathematics (inc Computing science) > QA299 Analysis, Calculus Central Services > Information ServicesCentral Services > Universities at MedwayCentral Services > Research and Innovation Services Bas Lemmens 28 Sep 2016 08:37 UTC 16 Feb 2021 13:37 UTC https://kar.kent.ac.uk/id/eprint/57539 (The current URI for this page, for reference purposes) https://orcid.org/0000-0001-6713-7683 https://orcid.org/0000-0002-8885-9156
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