Cohomological Characterizations of Biprojective and Biflat Banach Algebras

 Let A be a biprojective Banach algebra, and let A-mod-A be the category of Banach A-bimodules. In this paper, for every given -mod-A, we compute all the cohomology groups . Furthermore, we give some cohomological characterizations of biprojective Banach algebras. In particular, we show that the following properties of a Banach algebra A are equivalent to its biprojectivity: (i) for all -mod -A; (ii) for all -mod-A; (iii) for all -mod-A. (Here and are, respectively, the Banach A-bimodules of left, right and double multipliers of X.) Further, if A is a biflat Banach algebra and -mod-A, we compute all the cohomology groups , where is the Banach A-bimodule dual to X. Also, we give cohomological characterizations of biflat Banach algebras. We prove that a Banach algebra A is biflat if and only if any of the following conditions is valid: (i’) for all -mod-A; (ii’) for all -mod-A; (iii’) for all -mod-A. Received 16 June 1998     

Volterra-like Banach Algebras and their Second Duals

 This paper presents and studies a class of algebras which includes the usual Volterra algebra. Roughly speaking, they relate to the Volterra algebra in the way a general locally compact group relates to ℝ. We show that they can be viewed as quotients of some semigroup algebras introduced by Baker and Baker [1]. Their sets of nilpotent elements are dense. We investigate the second duals of these algebras and find that most of the properties found in [7] for the biduals of the group algebras L ¹(G) for compact G are retained here.     

Spectral synthesis for Banach algebras, II

This paper continues the study of spectral synthesis and the topologies and on the ideal space of a Banach algebra, concentrating particularly on the class of Haagerup tensor products of C-algebras. For this class, it is shown that spectral synthesis is equivalent to the Hausdorffness of . Under a weak extra condition, spectral synthesis is shown to be equivalent to the Hausdorffness of . Received: 6 October 1999; in final form: 15 May 2000 / Published online: 25 June 2001     

Gleason Parts and Weakly Compact Homomorphismsbetween Uniform Banach Algebras

 If points in nontrivial Gleason parts of a uniform Banach algebra have unique representing measures, then the weak and the norm topology coincide on the spectrum. We derive from this several consequences about weakly compact homomorphisms and discuss the case of other uniform Banach algebras arising in complex infinite dimensional analysis. (Received 9 March 1998; in revised form 29 June 1998)     

Universal Köthe Sequence Spaces

 A characterization is given for the K?the matrices B such that the K?the sequence space , with , contains all K?the sequence spaces of order p as subspaces. It follows that the class of K?the sequence spaces of order p has a universal element which is quasinormable. In particular, there is a quasinormable space (respectively, which contains every nuclear Fréchet space with basis (respectively, every countably normed Fréchet Schwartz space). Only Fréchet spaces with continuous norm are considered in this note. Received 15 January 1997; in final form 9 June 1997     

The Automorphism Group of a Compact Generalized Polygon has Finite Dimension

 For every compact generalized polygon whose point space has finite dimension, we show that the automorphism group has finite dimension as well. This opens the field for the application of Lie theoretic methods. Received 12 November 1997; in revised form 10 March 1998     

Involutorische Automorphismen ebener konvexer Mengen

 A bijection of the compact convex set with is called a reflection of K, if σ maps convex subsets of K into convex subsets. Conditions are stated unter which the existence of few reflections imply that is an ellipse. Received 4 March, 1998     

Topology on Asymptotic Algebras of Generalized Functions and Applications

 Starting from a locally convex metrisable topological space and from any asymptotic scale, we construct a generalized extension of this space. To those extensions, we associate Hausdorff topologies. We introduce the notion of a temperate map, with respect to a given asymptotic scale, between two locally convex metrisable semi-normed spaces. We show that such mappings extend in a canonical way to mappings between the respective generalized extensions. We give an application to nonlinear Dirichlet boundary value problems with singular data in the framework of generalized extensions. (Received 2 November 1998; in revised form 25 February 1999)     

On Pitt’s Theorem for Operators between Scalar and Vector-Valued Quasi-Banach Sequence Spaces

 We find natural conditions under which all continuous linear operators between two scalar or vector-valued quasi-Banach sequence spaces are compact. In the case of scalar-valued Banach sequence spaces we show that all such operators essentially factorize through diagonal operators between suitable -spaces. (Received 21 June 1999; in revised form 27 September 1999)     

Harmonic Functions on Homogeneous Spaces

 Given a locally compact group G acting on a locally compact space X and a probability measure σ on G, a real Borel function f on X is called σ-harmonic if it satisfies the convolution equation . We give conditions for the absence of nonconstant bounded harmonic functions. We show that, if G is a union of σ-admissible neighbourhoods of the identity, relative to X, then every bounded σ-harmonic function on X is constant. Consequently, for spread out σ, the bounded σ-harmonic functions are constant on each connected component of a [SIN]-group and, if G acts strictly transitively on a splittable metric space X, then the bounded σ-harmonic functions on X are constant which extends Furstenberg’s result for connected semisimple Lie groups. (Received 13 June 1998; in revised form 31 March 1999)     

Strongly τ-Decomposable and Selfdecomposable Laws on Simply Connected Nilpotent Lie Groups

 Let ? be a simply connected nilpotent Lie group with Lie Algebra ? and let τ be a contraction on ?. A probability measure μ on ? is strongly τ-decomposable iff it is representable as the limit of for some probability ν on ?. We show that such a limit exists if and only if ν possesses a finite logarithmic moment with respect to a homogeneous norm on ?. This result is then generalized to the class of selfdecomposable laws on ?. We also show that selfdecomposable laws on ? correspond in a 1–1 way to operator selfdecomposable laws on the tangent space ?. Received 1 October 1998; in revised form 29 March 1999     

Sum-Difference Sequences and Catalan Numbers

 Let be a sequence of natural numbers > 1, and set . The sequence is called admissible if a _i divides for all i. It is known that the admissible sequences are counted by the Catalan numbers. We present a proof of this fact which, in turn, leads to some interesting combinatorial and number-theoretic questions. Received 12 May 1997; in revised form 9 June 1997     

Pontryagin duality for topological Abelian groups

A topological Abelian group G is Pontryagin reflexive, or P-reflexive for short, if the natural homomorphism of G to its bidual group is a topological isomorphism. We look at the question, set by Kaplan in 1948, of characterizing the topological Abelian groups that are P-reflexive. Thus, we find some conditions on an arbitrary group G that are equivalent to the P-reflexivity of G and give an example that corrects a wrong statement appearing in previously existent characterizations of P-reflexive groups. Received: 10 February 2000 / Published online: 17 May 2001     

Banach Spaces Admitting a Separating Polynomial and L P Spaces

 We prove that in a Banach space admitting a separating polynomial, each weakly null normalized sequence has a subsequence which is equivalent to the usual basis of some , p an even integer. We show that for each even integer p, the Schatten class admits a separating polynomial. For a space with a basis admitting a 4-homogeneous separating polynomial, we relate the unconditionality of the basis with the existence of certain type of polynomials defined in terms of infinite symmetric matrices. We find relations between the properties of the entries of these matrices and the geometrical structure of the space. Finally we study the existence of convex 4-homogeneous separating polynomials. Received 3 January 2001     

On Patterns in Sequences of Random Events

 This paper considers patterns of particular events in sequences of trials, some independent and others Markovian. Matrix recursions are found for the number of sequences of length n avoiding a specific pattern, and the associated probability of this event is evaluated. A Markov chain method for the study of such problems is outlined, and is illustrated in various cases. Finally, configurations of length 3 in Bernoulli trials are examined as an example. Received 12 January 1998 in revised form 26 June 1998     

Constant Mean Curvature Surfaces with Boundaryin Hyperbolic Space

 We study constant mean curvature compact surfaces immersed in hyperbolic space with non-empty boundary (=H-surfaces). We prove that the only H-surfaces with boundary circular and 0≤∣H∣≤1, are the umbilical examples. When the surface is embedded, conditions to be umbilical are given. Finally, we characterize umbilical surfaces bounded by a circle among all H-discs with small area. Received 27 March 1997; in final form 11 June 1998     

On the Zero-Divergence of Equidistant Lagrange Interpolation

 In 1942, P. Szász published the surprising result that if a function f is of bounded variation on [−1, 1] and continuous at 0 then the sequence of the equidistant Lagrange interpolation polynomials converges at 0 to . In the present note we give a construction of a function continuous on [−1, 1] whose Lagrange polynomials diverge at 0. Moreover, we show that the rate of divergence attains almost the maximal possible rate. (Received 2 February 2000)