Homogeneous self dual cones versus Jordan algebras. The theory revisited

Bellissard, Jean; Iochum, B.

Annales de l'Institut Fourier, Tome 28 (1978) no. 1, pp. 27-67.

Soit $𝔐$ une J.B. algèbre, c’est-à-dire, une algèbre de Jordan-Banach dont la norme satisfait :

(i) $\parallel ab\parallel \le \parallel a\parallel \parallel b\parallel$

(ii) $\parallel a^2\parallel =\parallel a{\parallel }^2$

(iii) $\parallel a^2\parallel \le \parallel a^2+b^2\parallel$,    $a,b\in 𝔐$.

On suppose que $𝔐$ est monotone fermée (i.e., $𝔐$ coïncide avec ${𝔐}^{**}$) et que $(𝔐$ possède une trace finie, normale, fidèle. La fermeture $({\overline{𝔐}}^{+}{)}^{\varphi }$ de ${𝔐}^{+}=\{a^2\mid a\in 𝔐\}$ par rapport à la structure hilbertienne déduite de $\varphi$ est caractérisée par trois propriétés géométriques : autopolarité, homogénéité au sens de A. Connes, et existence d’un vecteur trace.

Let $𝔐$ be a Jordan-Banach algebra with identity 1, whose norm satisfies:

(i) $\parallel ab\parallel \le \parallel a\parallel \parallel b\parallel$,   $a,b\in 𝔐$

(ii) $\parallel a^2\parallel =\parallel a{\parallel }^2$

(iii) $\parallel a^2\parallel \le \parallel a^2+b^2\parallel$.

$𝔐$ is called a JB algebra (E.M. Alfsen, F.W. Shultz and E. Stormer, Oslo preprint (1976)). The set ${𝔐}^{+}$ of squares in $𝔐$ is a closed convex cone. $(𝔐,{𝔐}^{+},\mathbf{1})$ is a complete ordered vector space with $\mathbf{1}$ as a order unit. In addition, we assume $𝔐$ to be monotone complete (i.e. $𝔐$ coincides with the bidual ${𝔐}^{**}$), and that there exists a finite normal faithful trace $\varphi$ on $𝔐$.

Then the completion $\{{𝔐}^{+}{\}}_{\varphi }$ of ${𝔐}^{+}$ with respect to the Hilbert structure defined by $\varphi$, is characterized by three properties: self duality, homogeneity (in the sense of A. Connes, Ann. Inst. Fourier, Grenoble, 24, 4 (1974), 121–155) and existence of a trace vector.

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