Given a linear map T on Euclidean Jordan algebra of rank n, we consider the set all nonnegative vectors q in Rn with decreasing components that satisfy pointwise weak-majorization inequality λ(|T(x)|)≺w⁡q∗λ(|x|), where λ is eigenvalue and ∗ denotes componentwise product Rn. With respect to ordering, show existence least vector this set. When positive map, shown be join (in order) T(e) T∗(e), e ...

Let L be a linear transformation on a finite dimensional real Hilbert space H and K be a closed convex cone with dual K in H . The cone spectrum of L relative to K is the set of all real λ for which the linear complementarity problem x ∈ K, y = L(x)− λx ∈ K, and 〈x, y〉 = 0 admits a nonzero solution x. In the setting of a Euclidean Jordan algebra H and the corresponding symmetric cone K, we disc...

We describe the phase portrait of generalized aane-scaling vector elds for optimization problems involving symmetric cones. A Poisson structure on the complexiication of a real Jordan Euclidean algebra is introduced. Nonconstrained aane-scaling vector elds are proved to be Hamiltonian with respect to this Poisson structure. Constrained aane-scaling vector elds are obtained as a symplectic reduc...

In a recent paper, Chua and Yi introduced the so-called uniform nonsingularity property for a nonlinear transformation on a Euclidean Jordan algebra and showed that it implies the global uniqueness property in the context of symmetric cone complementarity problems. In a related paper, Chua, Lin, and Yi raise the question of converse. In this paper, we show that, for linear transformations, the ...

Jordan C*-algebras go back to Kaplansky, see [20]. Let J be a complex Banach Jordan algebra, that is, a complex Banach space with commutative bilinear product x◦y satisfying x◦(x2◦y) = x2◦(x◦y) as well as ||x◦y|| ≤ ||x||·||y||, Bounded symmetric domains and generalized operator algebras 51 and suppose that on J is given a (conjugate linear) isometric algebra involution x 7→ x∗. Then J is called...

The purpose of this paper is to introduce Hom-alternative algebras and Hom-Jordan algebras. We discuss some of their properties and provide construction procedures using ordinary alternative algebras or Jordan algebras. Also, we show that a polarization of Hom-associative algebra leads to Hom-Jordan algebra. INTRODUCTION Hom-algebraic structures are algebras where the identities defining the st...

We investigate the spectral geometry of the exceptional Jordan algebra and its extensions. We examine the spectrum of the exceptional Jordan algebra over the sixteen-dimensional space of primitive idempotents, where it exhibits three real eigenvalues. We interpret the spectrum as coordinates for a coincident D-brane system where the real eigenvalues correspond to positions of three D0-branes on...

We study a noncommutative generalization of Jordan algebras called Leibniz— Jordan algebras. These algebras satisfy the identities [x1x2]x3 = 0, (x 2 1 , x2, x3) = 2(x1, x2, x1x3), x1(x 2 1 x2) = x 2 1 (x1x2); they are related with Jordan algebras in the same way as Leibniz algebras are related to Lie algebras. We present an analogue of the Tits— Kantor—Koecher construction for Leibniz—Jordan a...

Let n ∈ N − {1}, and let A be a Banach algebra. An additive map D : A → A is called n-Jordan derivation if D(a) = D(a)a + aD(a)a + ...+ aD(a)a+ aD(a), for all a ∈ A. Using fixed point methods, we investigate the stability of n–Jordan derivations (n–Jordan ∗−derivations) on Banach algebras (C∗−algebras). Also we show that to each approximate ∗−Jordan derivation f in a C∗− algebra there correspon...

Let A be a unital algebra and M be a unital A-bimodule. A characterization of generalized derivations and generalized Jordan derivations from A into M, through zero products or zero Jordan products, is given. Suppose that M is a unital left A-module. It is investigated when a linear mapping from A into M is a Jordan left derivation under certain conditions. It is also studied whether an algebra...