C. Martín González

Gradings on the real forms of the Albert algebra, of g2, and of f4

We find all the fine group gradings on the real forms of the Albert algebra and of the exceptional Lie algebras g2 and f4.

The banach-lie group of lie triple automorphisms of an H*-algebra

Abstract We study the Banach-Lie group Ltaut(A) of Lie triple automorphisms of a complex associative H*-algebra A. Some consequences about its Lie algebra, the algebra of Lie triple derivations of A, Ltder(A), are obtained. For a topologically simple A, in the infinite-dimensional case we have Ltaut(A)0 = Aut(A) implying Ltder(A) = Der(A). In the finite-dimensional case Ltaut(A)0 is a direct product of Aut(A) and a certain subgroup of Lie derivations δ from A to its center, annihilating commutators.

Dual pairs techniques in H*-theories

This work is a version for Jordan pairs, of a previous result for Jordan algebras given in Rodriguez (1988). However, the tools we use are completely different from those in Rodriguez (1988). A Jordan H∗-pair is (in a sense) a complicated algebraic object enriched with a Hilbert space structure which is well related to its algebraic structure. In this work we describe a certain class of Jordan H∗-pairs by forgetting their Hilbert space structure and starting with the remaining purely algebraic information available on it. More precisely, if ((R+, R−), 〈 〉) is an associative pair such that ((R+, R−)J,) with x, y, z := 〈x, y, z〉 + 〈z, y, x〉 is a topologically simple Jordan H∗-pair, then R can be endowed of an (associative) H∗-pair structure such that its H∗-symmetrized agrees with the Jordan H∗-pair RJ.

A LINEAR APPROACH TO LIE TRIPLE AUTOMORPHISMS OF H*-ALGEBRAS

By developing a linear algebra program involving many different structures associated to a three-graded H*-algebra, it is shown that if L is a Lie triple automorphism of an infinite-dimensional topologically simple associative H*-algebra A, then L is either an automorphism, an anti-automorphism, the negative of an automorphism or the negative of an anti-automorphism. If A is finite-dimensional, then there exists an automorphism, an anti-automorphism, the negative of an automorphism or the negative of an anti-automorphism F : A A such that := F - L is a linear map from A onto its center sending commutators to zero. We also describe L in the case of having A zero annihilator.

Hilbert space methods in the theory of Lie triple systems

Abstract In [1] , Lister introduced the concept of Lie triple system and classified the finite-dimensional simple Lie triple system over an algebraically closed field of characteristic zero. However, the classification in the infinite-dimensional case is still an open problem. In order to study infinite-dimensional Lie triple systems, we introduce in this paper the notion of two-graded L*-algebra and L*-triple. We obtain a structure theory of infinite dimensional two-graded L* -algebras, and we also establish some results about L*-triples, as a classification of L*-triples admitting a two-graded L*-algebra envelope, their relation with L*-subtriples of A ‒ , for a ternary H*-algebra A, and the structure of direct limits of certain systems of L*-triples. As a tool, we develop a complete theory of direct limits of ternary H*-structures .

Rings whose class of projective modules is socle fine

A class C of modules over a unitary ring is said to be socle fine if whenever M, N ∈ C with Soc(M) ∼= Soc(N) then M ∼= N. In this work we characterize certain types of rings by requiring a suitable class of its modules to be socle fine. Then we study socle fine classes of quasi-injective, quasi-projective and quasicontinuous modules which we apply to find socle fine classes in special types of noetherian rings. We also initiate the study of those rings whose class of projective modules is socle fine.

Martindale Algebras of Quotients of Graded Algebras

The motivation for this paper is the study of the relation between the zero component of the maximal graded algebra of quotients and the maximal graded algebra of quotients of the zero component, both in the Lie case and when considering Martindale algebras of quotients in the associative setting. We apply our results to prove that the finitary complex Lie algebras are (graded) strongly nondegenerate and compute their maximal algebras of quotients.

Compact graph C ∗ -algebras

We show that compact graph C∗-algebras C∗(E) are topological direct sums of finite matrices over ℂ and KL(H), for some countably dimensional Hilbert space, and give a graph-theoretic characterization as those whose graphs are row-finite, acyclic and every infinite path ends in a sink. We further specialize in the simple case providing both structure and graph-theoretic characterizations. In order to reach our goals we make use of Leavitt path algebras Lℂ(E). Moreover, we describe the socle of C∗(E) as the two-sided ideal generated by the line point vertices.