We review various properties of the exceptional Euclidean Jordan algebra of degree three. Euclidean Jordan algebras of degree three and their corresponding Freudenthal triple systems were recently shown to be intimately related to extremal black holes in N = 2, d = 4 homogeneous supergravities. Using a novel type of eigenvalue problem with eigenmatrix solutions, we elucidate the rich matrix geo...

We study analyticity, differentiability, and semismoothness of Löwner’s operator and spectral functions under the framework of Euclidean Jordan algebras. In particular, we show that many optimization-related classical results in the symmetric matrix space can be generalized within this framework. For example, the metric projection operator over any symmetric cone defined in a Euclidean Jordan a...

In a recent article [8], Gowda and Sznajder studied the concept of Schur complement in Euclidean Jordan algebras and described Schur determinantal and Haynsworth inertia formulas. In this article, we establish some more results on the Schur complement. Specifically, we prove, in the setting of Euclidean Jordan algebras, an analogue of the Crabtree-Haynsworth quotient formula and show that any S...

In a recent paper [Linear Algebra Appl., 461:92--122, 2014], Tao et al. proved an analog of Thompson's triangle inequality for simple Euclidean Jordan algebra by using case-by-case analysis. this short note, we provide direct proof that is valid on any algebras.

In this paper, we give a characterization of strongly Jordan zero-product preserving maps on normed algebras as a generalization of  Jordan zero-product preserving maps. In this direction, we give some illustrative examples to show that the notions of strongly zero-product preserving maps and strongly Jordan zero-product preserving maps are completely different. Also, we prove that the direct p...

Let g be a continuously differentiable function whose derivative is matrix monotone on positive semi-axis. Such a function induces a function φ(x) = tr(g(x)) on the cone of squares of an arbitrary Euclidean Jordan algebra. We show that φ(x)− ln det(x) is a self-concordant function on the interior of the cone. We also show that − ln(t−φ(x))−ln det(x) is √ 5 3 (r+1)-self-concordant barrier on the...

Let Rα be the Riesz distribution on a simple Euclidean Jordan algebra, parametrized by α ∈ C. I give an elementary proof of the necessary and sufficient condition for Rα to be a locally finite complex measure (= complex Radon measure). Soit Rα la distribution de Riesz sur une algèbre de Jordan euclidienne simple, paramétrisée par α ∈ C. Je donne une démonstration élémentaire de la condition néc...

in this paper, we give a characterization of strongly jordan zero-product preserving maps on normed algebras as a generalization of  jordan zero-product preserving maps. in this direction, we give some illustrative examples to show that the notions of strongly zero-product preserving maps and strongly jordan zero-product preserving maps are completely different. also, we prove that the direct p...

We consider possible non-signaling composites of probabilistic models based on euclidean Jordan algebras. Subject to some reasonable constraints, we show that no such composite exists having the exceptional Jordan algebra as a direct summand. We then construct several dagger compact categories of such Jordan-algebraic models. One of these neatly unifies real, complex and quaternionic mixed-stat...