Separate weak*-continuity of the triple product in dual real JB*-triples

We prove that, if E is a real JB*-triple having a predual then is the unique predual of E and the triple product on E is separately $sigma (E,E_{*_{}})-$continuous. Received February 1, 1999; in final form March 29, 1999 / Published online May 8, 2000     

Classification of sequentially weakly continuous $JB^*$-triples

Let D be the open unit ball of a -triple A and let Aut(D) be the group of all biholomorphic automorphisms of D. It is shown that every element of Aut(D) is sequentially weakly continuous if and only if every primitive ideal of A is a maximal closed ideal and is a type I -triple without infinite-spin part. Implications for general structure theory are explored. In particular, it is deduced that every -triple A contains a smallest ideal J for which the sequentially weakly continuous biholomorphic automorphisms of the open unit ball of A/J are all linear. Received August 27, 1998; in final form February 10, 1999     

The characterizations of a semicircle law by the certain freeness in a C*-probability space

In usual probability theory, various characterizations of the Gaussian law have been obtained. For instance, independence of the sample mean and the sample variance of independently identically distributed random variables characterizes the Gaussian law and the property of remaining independent under rotations characterizes the Gaussian random variables. In this paper, we consider the free analogue of such a kind of characterizations replacing independence by freeness. We show that freeness of the certain pair of the linear form and the quadratic form in freely identically distributed noncommutative random variables, which covers the case for the sample mean and the sample variance, characterizes the semicircle law. Moreover we give the alternative proof for Nica's result that the property of remaining free under rotations characterizes a semicircular system. Our proof is more direct and straightforward one. Received: 12 February 1997 / Revised version: 16 June 1998     

Elementary operators on antiliminal C^*-algebras

We establish the equivalence of the following three properties of a -algebra A. (a) Every positive elementary operator on A is completely positive. (b) The norm and the cb-norm coincide for every elementary operator on A. (c) A is an extension of an antiliminal -algebra by an abelian one. Received: 15 July 1998 / in revised form: 22 September 1998     

A polynomial of graphs on surfaces

ribbon graphs , i.e., graphs realized as disks (vertices) joined together by strips (edges) glued to their boundaries, corresponding to neighbourhoods of graphs embedded into surfaces. We construct a four-variable polynomial invariant of these objects, the ribbon graph polynomial, which has all the main properties of the Tutte polynomial. Although the ribbon graph polynomial extends the Tutte polynomial, its definition is very different, and it depends on the topological structure in an essential way. Received: 14 September 2000 / Published online: 18 January 2002     

On summing operators on JB*-triples

In this paper we introduce 2-JB^*-triple-summing operators on real and complex JB^*-triples. These operators generalize 2-C^*-summing operators on C^*-algebras. We also obtain a Pietsch's factorization theorem in the setting of 2-JB^*-triple-summing operators on JB^*-triples. Received: 18 December 2000     

Little Grothendieck`s theorem for real JB*-triples

We prove that given a real JB*-triple E, and a real Hilbert space H, then the set of those bounded linear operators T from E to H, such that there exists a norm one functional and corresponding pre-Hilbertian semi-norm on E such that for all , is norm dense in the set of all bounded linear operators from E to H. As a tool for the above result, we show that if A is a JB-algebra and is a bounded linear operator then there exists a state such that for all . Received June 28, 1999; in final form January 28, 2000 / Published online March 12, 2001     

The projective rank of a Hermitian symmetric space: a geometric approach and consequences

The “Projective Rank” of a compact connected irreducible Hermitian symmetric space M has been defined as the maximal complex dimension of the compact totally geodesic complex submanifolds having positive holomorphic bisectional curvature with the induced K?hler metric. We present a geometric way to compute this invariant for the space M based on ideas developed in [1], [13] and [14]. As a consequence we obtain the following inequality relating the Projective Rank, Pr(M), the usual rank,rk(M), and the 2-number # (which is known to be equal to the Euler-Poincare characteristic in these spaces). Received: 6 June 2000 / Published online: 1 February 2002     

The generalized Hodge conjecture for the quadratic complex of lines in projective four–space

We verify the generalized Hodge conjecture GHC(X,5,2) for the quadratic complex of lines in projective four–space. Received: 27 February 1998 / Revised version: 13 May 1998     

Hypergraph connectivity augmentation

Received May 1995 / Revised version received May 1996 Published online March 16, 1999     

Unique tensor factorization of algebras

Tensor product decomposition of algebras is known to be non-unique in many cases. But, as will be shown here, a -indecomposable, finite-dimensional -algebra A has an essentially unique tensor factorization into non-trivial, -indecomposable factors . Thus the semiring of isomorphism classes of finite-dimensional -algebras is a polynomial semiring . Moreover, the field of complex numbers can be replaced by an arbitrary field of characteristic zero if one restricts oneself to schurian algebras. Received: 5 October 1998     

The projective rank of a Hermitian symmetric space: A geometric approach and consequences

The “Projective Rank” of a compact connected irreducible Hermitian symmetric space M has been defined as the maximal complex dimension of the compact totally geodesic complex submanifolds having positive holomorphic bisectional curvature with the induced K?hler metric. We present a geometric way to compute this invariant for the space M based on ideas developed in [1], [13] and [14]. As a consequence we obtain the following inequality relating the Projective Rank, the usual rank, and the 2-number (which is known to be equal to the Euler-Poincare characteristic in these spaces). Received: 6 June 2000 / Revised version: 6 August 2001 / Published online: 4 April 2002