Homogeneous self dual cones versus Jordan algebras. The theory revisited

Jean Bellissard; B. Iochum

Annales de l'institut Fourier (1978)

  • Volume: 28, Issue: 1, page 27-67
  • ISSN: 0373-0956

Abstract

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Let 𝔐 be a Jordan-Banach algebra with identity 1, whose norm satisfies:(i) ∥ a b ∥ ≤ ∥ 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.

How to cite

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Bellissard, Jean, and Iochum, B.. "Homogeneous self dual cones versus Jordan algebras. The theory revisited." Annales de l'institut Fourier 28.1 (1978): 27-67. <http://eudml.org/doc/74348>.

@article{Bellissard1978,
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.},
author = {Bellissard, Jean, Iochum, B.},
journal = {Annales de l'institut Fourier},
language = {eng},
number = {1},
pages = {27-67},
publisher = {Association des Annales de l'Institut Fourier},
title = {Homogeneous self dual cones versus Jordan algebras. The theory revisited},
url = {http://eudml.org/doc/74348},
volume = {28},
year = {1978},
}

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
PB - Association des Annales de l'Institut Fourier
VL - 28
IS - 1
SP - 27
EP - 67
AB - 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.
LA - eng
UR - http://eudml.org/doc/74348
ER -

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