Homogeneous self dual cones versus Jordan algebras. The theory revisited

Bellissard, Jean ; Iochum, B.

Annales de l'Institut Fourier, Tome 28 (1978), p. 27-67 / Harvested from Numdam

Résumé

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

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

(ii) $\Vert a^2\Vert = \Vert a\Vert ^2$

(iii) $\Vert a^2\Vert \le \Vert a^2 + b^2\Vert $,    $a,b \in {\frak M}$.

On suppose que ${\frak M}$ est monotone fermée (i.e., ${\frak M}$ coïncide avec ${\frak M}^{**}$) et que $({\frak M}$ possède une trace finie, normale, fidèle. La fermeture $(\overline{\frak M}^+)^\varphi $ de ${\frak M}^+ = \lbrace a^2\mid a\in {\frak M}\rbrace $ 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.

Abstract

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

(i) $\Vert ab\Vert \le \Vert a\Vert \ \Vert b\Vert $,   $a,b \in {\frak M}$

(ii) $\Vert a^2\Vert = \Vert a\Vert ^2$

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

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

Then the completion $\lbrace {\frak M}^+\rbrace _\varphi $ of ${\frak M}^+$ 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.

Détail

Publié le : 1978-01-01

MR 80b:46082 | Zbl 0365.46040 | 1 citation dans Numdam

DOI : https://doi.org/10.5802/aif.680

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