Grothendieck space ideals and weak continuity of polynomials on locally convex spaces

We introduce the classes of locally convex spaces with the local Dunford-Pettis property and locally dual Schur spaces. We examine their properties and their relationship to other classes of locally convex spaces. In the class of locally convex spaces with the local Dunford-Pettis property all polynomials are weakly sequentially continuous whereas in the class of locally dual Schur spaces all polynomials are weakly continuous on bounded sets. Research supported by Science Foundation Ireland, Basic Research Grant 2004.     

Invariant subsets of rank 1 manifolds

It is proved that for a Riemannian manifold M with nonpositive sectional curvature and finite volume the space of directions at each point in which geodesic rays avoid a sufficiently small neighborhood of a fixed rank 1 vector v∈UM looks very much like a generalized Sierpinski carpet. We also show for nonpositively curved manifolds M with dim M≥ 3 the existence of proper closed flow invariant subsets of the unit tangent bundle UM whose footpoint projection is the whole of M. Received: 6 July 2000 / Revised version: 11 October 2001     

Exact groups and Fell bundles

Kirchberg and Wasserman's Theorem, according to which a group is exact if and only if its reduced C*-algebra is exact, is extended to the context of Fell bundles. Received: 31 December 2000 / Revised version: 7 August 2001 / Published online: 4 April 2002     

Mixing and asymptotic properties of Markov semigroups on von Neumann algebras

In the paper, we investigate weak mixing and ‘approach to equilibrium’ of a Markov semigroup, i.e. a semigroup of linear normal positive unital mappings on a von Neumann algebra. In particular, we show that weak mixing is equivalent to ergodictity and triviality of the (point) spectrum of the semigroup, and give conditions assuring that each normal state tends to an equilibrium state under the action of the semigroup. Received May 6, 1999 / in final form October 21, 1999 / Published online July 20, 2000     

Compressions of free products of von Neumann algebras

A reduction formula for compressions of von Neumann algebra II–factors arising as free products is proved. This shows that the fundamental group is for some such algebras. Additionally, by taking a sort of free product with an unbounded semicircular element, continuous one parameter groups of trace scaling automorphisms on II–factors are constructed; this produces type III factors with core , where can be a full II–factor without the Haagerup approximation property. Received: 26 October 1998 / in final form 18 March 1999     

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     

A holomorphic characterization¶of C*- and JB*-algebras

We characterize C ^*-algebras as those complete normed associative complex algebras having approximate units bounded by one and whose open unit balls are bounded symmetric domains. Such a characterization follows from the more general fact, also proved in the paper, that non-commutative JB ^*-algebras coincide with complete normed (possibly non-associative) complex algebras having approximate units bounded by one and whose open unit balls are bounded symmetric domains. Received: 3 October 2000 / Revised version: 26 January 2001     

Cardinality of the set of extreme extensions¶of a quasi-measure

Cardinals that arise as the number of extreme quasi-measure extensions of a quasi-measure [resp., measure] μ defined on an algebra [resp., σ-algebra] of sets to a larger algebra [resp., σ-algebra] of sets are characterized in the general case as well as under some natural assumptions on μ. Received: 19 September 2000     

On a Problem of D. H. Lehmer and Kloosterman Sums

Let q3 be an odd number, a be any fixed positive integer with (a, q)=1. For each integer b with 1b<q and (b, q)=1, it is clear that there exists one and only one c with 0<c<q such that bca (mod q). Let N(a, q) denote the number of all solutions of the congruent equation bca (mod q) for 1b, c<q in which b and c are of opposite parity, and let . The main purpose of this paper is to study the distribution properties of E(a, q), and to give a sharper hybrid mean value formula involving E(a, q) and Kloosterman sums.Received January 24, 2002; in revised form August 12, 2002 Published online February 28, 2003     

Unbounded Disjointness Preserving Linear Functionals

Let X be a locally compact Hausdorff space and C ₀(X) the Banach space of continuous functions on X vanishing at infinity. In this paper, we shall study unbounded disjointness preserving linear functionals on C ₀(X). They arise from prime ideals of C ₀(X), and we translate it into the cozero set ideal setting. In particular, every unbounded disjointness preserving linear functional of c ₀ can be constructed explicitly through an ultrafilter on complementary to a cozero set ideal. This ultrafilter method can be extended to produce many, but in general not all, such functionals on C ₀(X) for arbitrary X. We also make some remarks where C ₀(X) is replaced by a non-commutative C*-algebra.     

A stratification of Hilbert schemes by initial ideals and applications

We show that the set of the homogeneous saturated ideals with given initial ideal in a fixed term-ordering is locally closed in the Hilbert scheme, and that it is affine if the initial ideal is saturated. Then, Hilbert schemes can be stratified using these subschemes. We investigate the behaviour of this stratification with respect to some properties of the closed points. As application, we describe the singular locus of the component of Hilb^{4 z +1} _ℙ4 containing the ACM curves of degree 4. Received: 30 November 1998 / Revised version: 16 September 1999     

Harder–Narasimhan reduction for principal bundles over a compact Kähler manifold

Generalizing the Harder–Narasimhan filtration of a vector bundle it is shown that a principal G-bundle over a compact K?hler manifold admits a canonical reduction of its structure group to a parabolic subgroup of G. Here G is a complex connected reductive algebraic group; in the special case where , this reduction is the Harder–Narasimhan filtration of the vector bundle associated to by the standard representation of . The reduction of in question is determined by two conditions. If P denotes the parabolic subgroup, L its Levi factor and the canonical reduction, then the first condition says that the principal L-bundle obtained by extending the structure group of the P-bundle using the natural projection of P to L is semistable. Denoting by the Lie algebra of the unipotent radical of P, the second condition says that for any irreducible P-module V occurring in , the associated vector bundle is of positive degree; here is considered as a P-module using the adjoint action. The second condition has an equivalent reformulation which says that for any nontrivial character of P which can be expressed as a nonnegative integral combination of simple roots (with respect to any Borel subgroup contained in P), the line bundle associated to for is of positive degree. The equivalence of these two conditions is a consequence of a representation theoretic result proved here. Received: 10 November 1999 / Revised version: 31 October 2001 / Published online: 26 April 2002     

Siegel modular forms and representations

This paper explicitly describes the procedure of associating an automorphic representation of PGSp(2n,?) with a Siegel modular form of degree n for the full modular group Γ _n =Sp(2n,ℤ), generalizing the well-known procedure for n=1. This will show that the so-called “standard” and ldquo;spinor”L-functions associated with such forms are obtained as Langlands L-functions. The theory of Euler products, developed by Langlands, applied to a Levi subgroup of the exceptional group of type F _<4, is then used to establish meromorphic continuation for the spinor L-function when n=3. Received: 28 March 2000 / Revised version: 25 October 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     

Operator space projective tensor product of $C^*$-algebras

For a finite dimensional -algebra A and any -algebra B, we determine a constant of equivalence of operator space projective norm and the Banach space projective norm on . We also discuss the *-Banach algebra . Received May 12, 1999; in final form September 8, 1999 / Published online April 12, 2001