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

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

(i) ∥ab∥≤∥a∥∥b∥,   a,b∈𝔐

(ii) ∥a 2 ∥=∥a∥ 2

(iii) ∥a 2 ∥≤∥a 2 +b 2 ∥.

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 + ,1) is a complete ordered vector space with 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 ϕ on 𝔐.

Then the completion {𝔐 + } ϕ of 𝔐 + with respect to the Hilbert structure defined by ϕ, 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) ∥ab∥≤∥a∥∥b∥

(ii) ∥a 2 ∥=∥a∥ 2

(iii) ∥a 2 ∥≤∥a 2 +b 2 ∥,    a,b∈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 (𝔐 ¯ + ) ϕ de 𝔐 + ={a 2 ∣a∈𝔐} par rapport à la structure hilbertienne déduite de ϕ est caractérisée par trois propriétés géométriques : autopolarité, homogénéité au sens de A. Connes, et existence d’un vecteur trace.

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     author = {Bellissard, Jean and Iochum, B.},
     title = {Homogeneous self dual cones versus {Jordan} algebras. {The} theory revisited},
     journal = {Annales de l'Institut Fourier},
     pages = {27--67},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {28},
     number = {1},
     year = {1978},
     doi = {10.5802/aif.680},
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Bellissard, Jean; Iochum, B. Homogeneous self dual cones versus Jordan algebras. The theory revisited. Annales de l'Institut Fourier, Tome 28 (1978) no. 1, pp. 27-67. doi: 10.5802/aif.680

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