Let be a Jordan-Banach algebra with identity 1, whose norm satisfies:
(i) ,
(ii)
(iii) .
is called a JB algebra (E.M. Alfsen, F.W. Shultz and E. Stormer, Oslo preprint (1976)). The set of squares in is a closed convex cone. is a complete ordered vector space with as a order unit. In addition, we assume to be monotone complete (i.e. coincides with the bidual ), and that there exists a finite normal faithful trace on .
Then the completion of with respect to the Hilbert structure defined by , 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 une J.B. algèbre, c’est-à-dire, une algèbre de Jordan-Banach dont la norme satisfait :
(i)
(ii)
(iii) , .
On suppose que est monotone fermée (i.e., coïncide avec ) et que possède une trace finie, normale, fidèle. La fermeture de par rapport à la structure hilbertienne déduite de 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}, pages = {27--67}, publisher = {Institut Fourier}, address = {Grenoble}, volume = {28}, number = {1}, year = {1978}, doi = {10.5802/aif.680}, zbl = {0365.46040}, mrnumber = {80b:46082}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.680/} }
TY - JOUR AU - Bellissard, Jean AU - Iochum, B. TI - Homogeneous self dual cones versus Jordan algebras. The theory revisited JO - Annales de l'Institut Fourier PY - 1978 SP - 27 EP - 67 VL - 28 IS - 1 PB - Institut Fourier PP - Grenoble UR - https://aif.centre-mersenne.org/articles/10.5802/aif.680/ DO - 10.5802/aif.680 LA - en ID - AIF_1978__28_1_27_0 ER -
%0 Journal Article %A Bellissard, Jean %A Iochum, B. %T Homogeneous self dual cones versus Jordan algebras. The theory revisited %J Annales de l'Institut Fourier %D 1978 %P 27-67 %V 28 %N 1 %I Institut Fourier %C Grenoble %U https://aif.centre-mersenne.org/articles/10.5802/aif.680/ %R 10.5802/aif.680 %G en %F 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. https://aif.centre-mersenne.org/articles/10.5802/aif.680/
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