Homogeneous self dual cones versus Jordan algebras. The theory revisited
Annales de l'Institut Fourier, Volume 28 (1978) no. 1, p. 27-67

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

(i) $\parallel ab\parallel \le \parallel a\parallel \phantom{\rule{4pt}{0ex}}\parallel b\parallel$,   $a,b\in 𝔐$

(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. $\left(𝔐,{𝔐}^{+},\mathbf{1}\right)$ 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 $\left\{{𝔐}^{+}{\right\}}_{\varphi }$ 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.

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

(i) $\parallel ab\parallel \le \parallel a\parallel \phantom{\rule{4pt}{0ex}}\parallel b\parallel$

(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 $\left(𝔐$ possède une trace finie, normale, fidèle. La fermeture $\left({\overline{𝔐}}^{+}{\right)}^{\varphi }$ de ${𝔐}^{+}=\left\{{a}^{2}\mid a\in 𝔐\right\}$ 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.

@article{AIF_1978__28_1_27_0,
author = {Bellissard, Jean and Iochum, B.},
title = {Homogeneous self dual cones versus Jordan algebras. The theory revisited},
journal = {Annales de l'Institut Fourier},
publisher = {Imprimerie Louis-Jean},
volume = {28},
number = {1},
year = {1978},
pages = {27-67},
doi = {10.5802/aif.680},
zbl = {0365.46040},
mrnumber = {80b:46082},
language = {en},
url = {http://www.numdam.org/item/AIF_1978__28_1_27_0}
}

Bellissard, Jean; Iochum, B. Homogeneous self dual cones versus Jordan algebras. The theory revisited. Annales de l'Institut Fourier, Volume 28 (1978) no. 1, pp. 27-67. doi : 10.5802/aif.680. http://www.numdam.org/item/AIF_1978__28_1_27_0/

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