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Mathematics > Functional Analysis

Title: An order theoretic characterization of spin factors

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 $g\colon C^\circ\to C^\circ$ with the property that $g$ is antihomogeneous, i.e., $g(\lambda x) =\lambda^{-1}g(x)$ for all $\lambda>0$ and $x\in C^\circ$, and $g$ is an order-antimorphism, i.e., $x\leq_C y$ if and only if $g(y)\leq_C g(x)$. 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 $\dim V\geq 3$, then there exists a bijective antihomogeneous order-antimorphism $g\colon C^\circ\to C^\circ$ if and only if $(V,C,u)$ is a spin factor.
Comments: 15 pages
Subjects: Functional Analysis (math.FA); Operator Algebras (math.OA)
MSC classes: 17C65, 46B40
Cite as: arXiv:1609.08304 [math.FA]
  (or arXiv:1609.08304v1 [math.FA] for this version)

Submission history

From: Bas Lemmens [view email]
[v1] Tue, 27 Sep 2016 08:14:29 GMT (15kb)