Alberto Castellón Serrano

Strongly prime alternative pairs with minimal inner ideals

In this paper, we use the known classification of the finite capacity simple alternative pairs and the version of the Litoff Theorem for Jordan pairs to describe all the strongly prime alternative pairs with nonzero socle. We study the inheritance of some properties (primeness, nondegenerancy,…) when passing from the original alternative pair to the symmetrized pair. Thus, we can apply Jordan theoretical results to the alternative case.

Special jordan H*-triple systems

This work, jointly with [9], completes the structure theory and classification of the Jordan H *- triple systems. The problem of describing the Jordan H *-triple systems is reduced in [5] to that of describing the topologically simple ones. Ruling out the finite-dimensional case, we have that any of these H *-triples has an underlying triple system structure of quadratic type (and these can be fully described), or it is the H *-triple system associated to the odd part of a topologically simple Z2-graded Jordan H *-algebra, whose classification is given in [13].

Prime Alternative Triple Systems

In [8] Loos obtains a structure theorem for finite dimensional simple alternative triple systems over an algebraically closed field of characteristic ≠ 2. This theorem has been generalized in [10] (resp. [9]) for simple alternative triple systems (resp. alternative pairs) containing a maximal tripotent (resp. a maximal idempotent). Other classes of alternative triple systems without maximal tripotent over a Hilbert space (real or complex) have been characterized in [2] and [3] where the simplicity is replaced by the topological simplicity and other hypothesis involving an involution and the inner product. In this paper, we obtain a description of nondegenerate prime alternative triple systems containing a maximal tripotent in terms of its central closure. A socle theory for prime nondegenerate alternative triple systems with maximal tripotent is established in a forthcoming paper [4].

Jordan H*-triple systems

This work, jointly with [7], gives a complete classification of Jordan H*-triple systems. There, the infinite—dimensional topologically simple special nonquadratic Jordan H*-triple systems are fully described in terms of the odd part of a ℤ2—graded H*—algebra. Here we complete the structure theory endowing to any simple finite—dimensional real Jordan triple system, of an H*—structure, essentialy unique, and determining the ones of quadratic type.