Algebras generated by inner functions

Let A be a uniform algebra on a compact space X. An inner function is a function in A unimodular on X. For three algebras of type H^∞ we prove A is generated by its inner functions. Whenever A is generated by its inner functions we prove the unit ball of A is the closed convex ball of the inner functions.     

On algebras generated by convolutions and discontinuous functions

Banach algebras generated by Fourier and Mellin convolution operators with discontinuous presymbols and by discontinuous functions in L_p (IR⁺, x) spaces with weight are investigated. The Fredholm properties are characterized by a symbol calculus and an index formula for such operators is presented. These results were obtained by H. O. Cordes in [3] for the case p=2, =0 and presymbols, which are discontinuous only at infinity and generalized in [20] for 1相似文献   

Algebraic functions in varieties generated by lattice-primal algebras

It is shown that in every algebra from a variety generated by a lattice-primal algebra, the algebraic functions are exactly those preserving all quasiorders of a certain type (the so-called admissible quasiorders).To the memory of András HuhnPresented by Alden Pixley.     

Convergence generated by analytic functionals,and isomorphism of algebras of analytic functions

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 40, No. 6, pp. 799–803, November–December, 1988.     

Representations of algebras in varieties generated by infinite primal algebras

If \({\mathcal{A}}\) is an infinite primal algebra, then we shall represent any algebra in the variety \({V\,(\mathcal{A}}\) ) generated by \({\mathcal{A}}\) as a limit reduced power of \({\mathcal{A}}\) . Furthermore, we show that any homomorphism between algebras in \({V\,(\mathcal{A}}\) ) can be induced by mappings between underlying sets of the limit reduced powers. With this representation of the morphisms between algebras in \({V\,(\mathcal{A}}\) ) at hand, we will construct a category equivalent to the category \({V\,(\mathcal{A}}\) ).     

Chordal Loewner families and univalent Cauchy transforms

We study chordal Loewner families in the upper half-plane and show that they have a parametric representation. We show one, that to every chordal Loewner family there corresponds a unique measurable family of probability measures on the real line, and two, that to every measurable family of probability measures on the real line there corresponds a unique chordal Loewner family. In both cases the correspondence is being given by solving the chordal Loewner equation. We use this to show that any probability measure on the real line with finite variance and mean zero has univalent Cauchy transform if and only if it belongs to some chordal Loewner family. If the probability measure has compact support we give two further necessary and sufficient conditions for the univalence of the Cauchy transform, the first in terms of the transfinite diameter of the complement of the image domain of the reciprocal Cauchy transform, and the second in terms of moment inequalities corresponding to the Grunsky inequalities.     

The variety generated by equivalence algebras

Every equivalence relation can be made into a groupoid with the same underlying set if we define the multiplication as follows: xy = x if x,y are related; otherwise, xy = y. The groupoids, obtained in this way, are called equivalence algebras. We find a finite base for the equations of equivalence algebras. The base consists of equations in four variables, and we prove that there is no base consisting of equations in three variables only. We also prove that all subdirectly irreducibles in the variety generated by equivalence algebras are embeddable into the three-element equivalence algebra, corresponding to the equivalence with two blocks on three elements. Received September 21, 1998; accepted in final form May 11, 1999.     

On algebras generated by positive operators

We study algebras generated by positive matrices, i.e., matrices with nonnegative entries. Some of our results hold in more general setting of vector lattices. We reprove and extend some theorems that have been recently shown by Kandi? and ?ivic. In particular, we give a more transparent proof of their result that the unital algebra generated by positive idempotent matrices E and F such that \(E F \ge F E\) is equal to the linear span of the set \(\{I, E, F, E F, F E, E F E, F E F, (E F)^2, (F E)^2\}\), and so its dimension is at most 9. We give examples of two positive idempotent matrices that generate unital algebra of dimension 2n if n is even, and of dimension \((2n - 1)\) if n is odd. We also prove that the algebra generated by positive matrices \(B_1\), \(B_2, \ldots , B_k\) is triangularizable if \(A B_i \ge B_i A\) (\(i=1,2, \ldots , k\)) for some positive matrix A with distinct eigenvalues.     

Lie algebras generated by 3-forms

Let U be a real vector space, B an inner product on U and $T\in \wedge U^{*}$ a 3-form. The 3-form T defines two natural maps, ${[?,?]}_U:{\wedge }^{2}U\to U$ and $\sigma :U\to {\wedge }^2{U}^{*}?\mathrm{so}(U,B)$ given by ${[x,y]}_U=2B^{?}(T(x,y,?))$ and $\sigma (x)=T(x,?,?)$. We show that ${[?,?]}_U$ is a Lie bracket if and only if $g_T\equiv \mathrm{Im}(\sigma )$ is a Lie subalgebra of $\mathrm{so}(U,B)$. To cite this article: R.P. Rohr, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Equational classes generated by finite algebras

Algebra universalis -     

Orthonormal bases generated by Cuntz algebras

We show how some orthonormal bases can be generated by representations of the Cuntz algebra; these include Fourier bases on fractal measures, generalized Walsh bases on the unit interval and piecewise exponential bases on the middle third Cantor set.     

On the numerical evaluation of Cauchy transforms

In this paper we consider simple methods for the reconstruction of the Cauchy transform over a curve when an explicit parametrization of the latter is not provided. The methods consist of replacing the parametrization of the curve by piecewise polynomial interpolation followed by the use of Newton-Cotes type formulae for the integration. The order of convergence of the resulting quadrature is higher than would be expected on the basis of considerations involving just interpolation theory, provided that the Cauchy transform is evaluated at known nodes on the curve. These results allow the calculation of the Cauchy transform at other points with the same accuracy if this scheme is followed by an interpolatory formula of sufficiently high accuracy.     

Operator algebras generated by Boolean algebras of projections in Montel spaces

It is shown that the classical result of W. Bade, stating that the uniformly closed operator algebra generated by a complete Boolean algebra of projections in a Banach space coincides with the weak operator closed algebra that it generates, is also valid (if suitably interpreted) in Montel spaces, Schwartz spaces and other related spaces.     

Images of non-tangential sectors under Cauchy transforms

Let μ be a complex measure on the circle. We denote byPμ andQμ the Poisson and conjugate Poisson integrals of μ in the unit disk, respectively. In this note, we study the relative asymptotic growth ofPμ andQμ near the support of μ. In particular, we show that on μ-almost every radius,Qμ grows no faster thanPμ. We also discuss related questions on the boundary behavior of analytic functions. The author is supported in part by N.S.F. grant DMS 9970151.