José Antonio Cuenca Mira

On one-sided division infinite-dimensional normed real algebras

In this note we introduce the concept of Cayley homomorphism which is closely related with those of composition algebra and normalized orthogonal multiplication. The key result shows the existence of certain types of Cayley homomorphisms for infinite dimension. As an application we prove the existence of left division infinite-dimensional complete normed real algebras with left unity.

Isomorphisms of H* -algebras

H *-algebras were introduced and studied by Ambrose [1] in the associative case, and the theory has been extended to such particular classes of non-associative algebras as Lie [18, 19], Jordan[20, 21, 7], alternative [11] and non-commutative Jordan [6] algebras. In all these cases the core of the matter is showing that every H *-algebra (in the given class) with zero annihilator is the closure of the orthogonal sum of its minimal closed ideals (each of which is a topologically simple H *-algebra), and then listing all the topologically simple H *-algebras in the class. In fact every nonassociative H *-algebra with zero annihilator is the closure of the orthogonal sum of its minimal closed ideals [6, theorem 2·7], so the problem of the classification of topologically simple non-associative H *-algebras becomes interesting. In relation with this problem the question arises whether, once an algebra A has been structured as a topologically simple H *-algebra, every H *-algebra structure on A is (up to a positive multiple of the inner product) totally isomorphic to the given one (see [3] and [11, section 4]). As a consequence of the results in this paper we give a general affirmative answer to this question.

On composition and absolute-valued algebras

In this note we give a complete description of the composition algebras A over fields of characteristic ≠ 2, 3 in the following cases: if A has an anisotropic norm and x 2 x = xxx 2 for every element; when A has a unitary central idempotent, it satisfies the identity ( x 2 x 2 ) x 2 = x 2 ( x 2 x 2 ), and A is of finite dimension or has anisotropic norm. As a consequence, we obtain the existence, up to an isomorphism, of only seven absolute-valued algebras with a non-zero central idempotent where the last identity holds. This result completes the study of the absolute-valued algebras of this kind that was initiated by El-Mallah and Agawany. We also introduce the class of e -quadratic algebra, which contains the quadratic algebras, but also includes large classes of composition and absolute-valued algebras. Many results on composition, absolute-valued and e -quadratic algebras are shown, and new proofs of some well-known theorems are given.

Third-power associative absolute valued algebras with a nonzero idempotent commuting with all idempotents

This paper deals with the determination of the absolute valued algebras with a nonzero idempotent commuting with the remaining idempotents and satisfying x2x = xx2 for every x. We prove that, in addition to the absolute valued algebras R, C, H, or O of the reals, complexes, division real quaternions or division real octonions, one such absolute valued algebra A can also be isometrically isomorphic to some of the absolute valued algebras C. H or O, obtained from C, H, and O by imposing a new product defined by multiplying the conjugates of the elements. In particular, every absolute valued algebra having the above properties is finite-dimensional. This generalizes some well known theorems of Albert, Urbanik and Wright, and El-Mallah.

ON DIVISION ALGEBRAS SATISFYING MOUFANG IDENTITIES

ABSTRACT Among the algebraic identities satisfied by alternative algebras play a fundamental role the so called Moufang identities. In the first section of this note we characterize division algebras satisfying one of the sided Moufang identities. As a consequence we obtain all the division normed real Moufang sided algebras. Up to isomorphism there are seven algebras in both cases, including the well known alternative algebras , , and . Moreover, there are three division right Moufang algebras with dimensions 2, 4 and 8 and another three division left Moufang algebras of the same dimensions. Finally we prove that , , and are the only division normed algebras satisfying the Moufang middle identity.ABSTRACT Among the algebraic identities satisfied by alternative algebras play a fundamental role the so called Moufang identities. In the first section of this note we characterize division algebras satisfying one of the sided Moufang identities. As a consequence we obtain all the division normed real Moufang sided algebras. Up to isomorphism there are seven algebras in both cases, including the well known alternative algebras , , and . Moreover, there are three division right Moufang algebras with dimensions 2, 4 and 8 and another three division left Moufang algebras of the same dimensions. Finally we prove that , , and are the only division normed algebras satisfying the Moufang middle identity.

COMPOSITION ALGEBRAS SATISFYING MOUFANG IDENTITIES

ABSTRACT This note is strongly inspired by[1] and[2] and it is devoted to the determination of all the composition algebras satisfying some of the Moufang identities.ABSTRACT This note is strongly inspired by[1] and[2] and it is devoted to the determination of all the composition algebras satisfying some of the Moufang identities.

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].

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.

On the Structure of Third-Power Associative Absolute Valued Algebras

This paper deals with pairs of nonzero idempotents e and f of a third-power associative absolute valued algebra A satisfying