Homogeneous self dual cones versus Jordan algebras. The theory revisited

Abstract

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.