Angel Rodríguez Palacios

On real forms of JB*-triples

We introduce real JB*-triples as real forms of (complex) JB*-triples and give an algebraic characterization of surjective linear isometries between them. As main result we show: A bijective (not necessarily continuous) linear mapping between two real JB*-triples is an isometry if and only if it commutes with the cube mappinga→a3={aaa}. This generalizes a result of Dang for complex JB*-triples. We also associate to every tripotent (i.e. fixed point of the cube mapping) and hence in particular to every extreme point of the unit ball in a real JB*-triple numerical invariants that are respected by surjective linear isometries.

The uniqueness of the complete norm topology in complete normed nonassociative algebras

Abstract A theorem on uniqueness of the complete norm topology for complete normed nonassociative algebras is proved. This theorem contains the well-known one by Johnson for associative Banach algebras and the recent analogous result by Aupetit for Banach-Jordan algebras.

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.

Jordan axioms for C*-algebras

A complex Banach spaceA which is also an associative algebra provided with a conjugate linear vector space involution * satisfying (a2)*=(a*)2, ∥aa*a∥=∥a∥3 and ∥ab+ba∥≦2∥a∥∥b∥ for alla, b inA is shown to be a C*-algebra. The assumptions onA can be expressed in terms of the Jordan algebra obtained by symmetrization of the product ofA and are satisfied by any C*-algebra. Thus we obtain a purely Jordan characterization of C*-algebras.A complex Banach spaceA which is also an associative algebra provided with a conjugate linear vector space involution * satisfying (a2)*=(a*)2, ∥aa*a∥=∥a∥3 and ∥ab+ba∥≦2∥a∥∥b∥ for alla, b inA is shown to be a C*-algebra. The assumptions onA can be expressed in terms of the Jordan algebra obtained by symmetrization of the product ofA and are satisfied by any C*-algebra. Thus we obtain a purely Jordan characterization of C*-algebras.

Absolute Valued Algebras of Degree Two

Following [2], we will say that a (nonassociative) algebra A is algebraic if, for every x in A, the subalgebra A(x) of A generated by x is finite-dimensional. If in fact dim(A(x)) ≤ m for all x in A and some natural number m only depending on A, then the algebraic algebra A is called of bounded degree, and the smallest such a number m is called the (bounded) degree of A. From the fact that C is the only finite-dimensional absolute valued complex algebra it follows easily that also C is the only algebraic absolute valued complex algebra. In the same way, R is the only absolute valued real algebra of degree one.

Absolute Valued Algebras with Involution

We study absolute valued algebras with involution, as defined in Urbanik (1961). We prove that these algebras are finite-dimensional whenever they satisfy the identity (x, x 2, x) = 0, where (·, ·, ·) means associator. We show that, in dimension different from two, isomorphisms between absolute valued algebras with involution are in fact *-isomorphisms. Finally, we give a classification up to isomorphisms of all finite-dimensional absolute valued algebras with involution. As in the case of a parallel situation considered in Rochdi (2003), the triviality of the group of automorphisms of such an algebra can happen in dimension 8, and is equivalent to the nonexistence of 4-dimensional subalgebras.

Isometries of JB-algebras

A bijective linear mapping between two JB-algebrasA andB is an isometry if and only if it commutes with the Jordan triple products ofA andB. Other algebraic characterizations of isometries between JB-algebras are given. Derivations on a JB-algebraA are those bounded linear operators onA with zero numerical range. For JB-algebras of selfadjoint operators we have: IfH andK are left Hilbert spaces of dimension ≥3 over the same fieldF (the real, complex, or quaternion numbers), then every surjective real linear isometryf fromS(H) ontoS(K) is of the formf(x)=UoxoU−1 forx inS(H), whereτ is a real-linear automorphism ofF andU is a real linear isometry fromH ontoK withU(λh)=τ(λ)U(h) for λ inF andh inH.

On the socle of a noncommutative Jordan algebra

In this paper we study some questions related to the socle of a nondegenerate noncommutative Jordan algebra. First we show that elements of finite rank belong to the socle, and that every element in the socle is von Neumann regular and has finite spectrum. Next we show that for Jordan Banach algebras the socle coincides with the maximal von Neumann regular ideal. For a nondegenerate noncommutative Jordan algebra, the annihilator of its socle can be regarded as a radical which is, generally, larger than Jacobson radical. Moreover, a nondegenerate noncommutative Jordan algebra whose socle has zero annihilator is isomorphic to a subdirect sum of primitive algebras having nonzero socle (which were described in [4]). Finally, these results are specialized to the particular case of an alternative algebra.In this paper we study some questions related to the socle of a nondegenerate noncommutative Jordan algebra. First we show that elements of finite rank belong to the socle, and that every element in the socle is von Neumann regular and has finite spectrum. Next we show that for Jordan Banach algebras the socle coincides with the maximal von Neumann regular ideal. For a nondegenerate noncommutative Jordan algebra, the annihilator of its socle can be regarded as a radical which is, generally, larger than Jacobson radical. Moreover, a nondegenerate noncommutative Jordan algebra whose socle has zero annihilator is isomorphic to a subdirect sum of primitive algebras having nonzero socle (which were described in [4]). Finally, these results are specialized to the particular case of an alternative algebra.

Non associative C*-algebras revisited

We give a detailed survey of some recent developments of non-associative C*-algebras. Moreover, we prove new results concerning multipliers and isometries of non-associative C*-algebras.

Bounded degree of weakly algebraic topological Lie algebras

We prove in this paper that a weakly algebraic Lie algebraA which is also a topological Baire algebra over a complete non-discrete valuated fieldK must be in fact algebraic of bounded degree. Similar results are also proven forp-algebraic restricted Liep-algebras, and for algebraic non-associative algebras.