Fréchet algebras of finite type

The main objects of study in this paper are Fréchet algebras having an Arens Michael representation in which every Banach algebra is finite dimensional. We shall classify these algebras using a theorem which ensures that the image of any continuous linear map of a Fréchet space of finite type (i.e., for which the defining seminorms have a finite dimensional cokernel) into any Fréchet space is in fact closed.This work is part of the research project of the European Research Training Network Analysis and Operators, contract HPRN-CT 2000 00116, funded by the European Commission.     

The bitangential inverse spectral problem for canonical systems

The theory of the direct and bitangential inverse input impedance problem is used to solve the direct and bitangential inverse spectral problem. The analysis of the direct spectral problem uses and extends a number of results that appear in the literature. Special attention is paid to the class of canonical integral systems with matrizants that are strongly regular J-inner matrix valued functions in the sense introduced in [7]. The bitangential inverse spectral problem is solved in this class. In our considerations, the data for this inverse problem is a given nondecreasing p×p matrix valued function σ(μ) on and a normalized monotonic continuous chain of pairs , of entire inner p×p matrix valued functions. Each such chain defines a class of canonical integral systems in which we find a solution of the inverse problem for the given spectral function σ(μ). A detailed comparison of our investigations of inverse problems with those of Sakhnovich is presented.     

Convex sets which are intersections of closed balls

In this paper we aim to investigate different questions concerning the stability of the set of all intersections of closed balls in a normed space. We are mainly concerned with: (i) the stability of under the closure of the vector sums; (ii) the stability under the addition of balls. We prove that (i) and (ii) are different properties which have strong connections with the geometry of the space. They have interest both in finite and infinite dimension. In the former case, there is a link with linear programming theory. We also study two more stability properties related to the well-known binary intersection property. Mazur sets and Mazur spaces are introduced, as a natural family satisfying (i). We prove that every two-dimensional normed space is a Mazur space, a result which distinguishes dimension d?2 from dimension d?3. We also discuss the connections between Mazur spaces and porosity.     

Beurling’s Phenomenon in Two Variables

In the Hardy space over the unit disk H²(D), every shift-invariant subspace M is of the form H²(D)) for some inner function by Beurlings theorem, and the reproducing kernel of M is . The fact that is inner implies that is subharmonic and has boundary value 1 almost everywhere on T. In the two variable space H²(D²), things are far more complicated and there is no similar characterization of invariant subspaces M in terms of inner functions. However, we will show in this paper an analogous phenomenon in terms of reproducing kernels, namely, is subharmonic and has boundary value 1 almost everywhere on T². The proof uses an index theorem obtained recently.     

Bounded Reflexivity of Operator Spaces

We introduce a concept bounded reflexivity for a subspace of operators on a normed space. We explore the properties of bounded reflexivity, study the similarities and differences between bounded reflexivity and the usual reflexivity for a subspace of operators. As applications of bounded reflexivity, we give alternative proofs of some well known results about positivity and complete positivity of elementary operators.     

Regularity of Integral Operators

In this paper we consider a class of integral operators whose kernels satisfy certain generalizations of Hörmander condition and establish their regularity results.     

Singly generated planar algebras of small dimension, Part II

We classified in Bisch and Jones (Duke Math. J. 101 (2000) 41) all spherical C∗-planar algebras generated by a non-trivial 2-box subject to the condition that the dimension of N′∩M₂ is ?12. We showed that they are given by the Fuss-Catalan systems discovered in Bisch and Jones (Invent. Math. 128 (1997) 89) and one exceptional planar algebra. In the present paper, we extend these results and show that there is only one spherical C∗-planar algebra generated by a single non-trivial 2-box if the dimension of N′∩M₂ is 13. It is given by the standard invariant of the crossed product subfactor , where D₅ denotes the dihedral group with 10 elements.     

Operators of Fourier type <Emphasis Type="Italic">p</Emphasis> with respect to some subgroups of a locally compact Abelian group

It is shown that all Banach space operators which have Fourier type p(1 < p 2) with respect to a second countable locally compact abelian group G also have Fourier type p with respect to every closed discrete subgroup H of G. The same statement holds for any closed subgroup H of G when p = 2. Also shown as a corollary is that Fourier type 2 operators have Walsh type 2. Received: 10 June 2002     

The central kernel of the Peirce-one space

Further investigation into the properties of the Peirce-one space J₁ corresponding to a weak^*-closed inner ideal J in a JBW^*-triple A is carried out, and, in particular, it is shown that J₁ contains no non-trivial weak^*-closed ideals.Received: 12 June 2002     

Extension and lifting of polynomials

We characterize the Banach spaces enjoying polynomial lifting properties, and the spaces that admit compact extensions of polynomials.Received: 20 June 2002