Using Bessel-Muirhead system, we can express the K-bessel function defined on a Jordan algebra as linear combination of the J-solutions. We determine explicitly the coefficients when the rank of this Jordan algebra is three after a reduction to the rank two. The main tools are some algebraic identities developed for the occasion.

We show that a Jordan algebra of compact quasinilpotent operators which contains a nonzero trace class operator has a common invariant subspace. As a consequence of this result, we obtain that a Jordan algebra of quasinilpotent Schatten operators is simultaneously triangularizable.

This article deals with necessary and sufficient conditions for a family of elements in Euclidean Jordan algebra to have simultaneous (order) spectral decomposition. Motivated by well-known matrix theory result that any pairwise commuting complex Hermitian matrices is simultaneously (unitarily) diagonalizable, we show the setting general algebra, operator has decomposition, i.e. there exists co...

in this paper, lie group structure and lie algebra structure of unit complex 3-sphere     are studied. in order to do this, adjoint representations of unit biquaternions (complexified quaternions) are obtained. also, a correspondence between the elements of     and the special complex unitary matrices    (2) is given by expressing biquaternions as 2-dimensional bicomplex numbers    .  the relat...

There have been several propositions for a geometric and essentially non-linear formulation of quantum mechanics, see, e.g., [AS98], [BH01], [CGM03], [Ki79]. From a purely mathematical point of view, the point of view of Jordan algebra theory might give new strength to such approaches: there is a “Jordan geometry” belonging to the Jordan part of the algebra of observables, in the same way as Li...

A recent paper by N. Jacobson' develops a theory of Jordan algebras with minimum condition which is similar to the associative theory of semisimple Artinian rings. Jacobson shows that any algebra satisfying his axioms is a direct sum of a finite number of simple algebras satisfying his axioms, and he deduces the structure of those simple algebras which contain three mutually orthogonal idempote...

In this note I discuss some aspects of a formulation of quantum mechanics based entirely on the Jordan algebra of observables. After reviewing some facts of the formulation in the C∗-approach I present a Jordan-algebraic Hilbert space construction (inspired by the usual GNS-construction), thereby obtaining a real Hilbert space and a (Jordan-) representation of the algebra of observables on this...