Ball Algebra Research Articles

Unified Approach to Spectral Properties of Multipliers

Let $\mathbb{B}_n$ be the open unit ball in $\mathbb{C}^n$. We characterize the spectra of pointwise multipliers $M_u$ acting on Banach spaces of analytic functions on $\mathbb{B}_n$ satisfying some general conditions. These spaces include Bergman-Sobolev spaces $A^p_{\alpha,\beta}$, Bloch-type spaces $\mathcal{B}_{\alpha}$, weighted Hardy spaces $H^p_w$ with Muckenhoupt weights and Hardy-Sobolev Hilbert spaces $H^2_{\beta}$. Moreover, we describe the essential spectra of multipliers in most of the aforementioned spaces, in particular, in those spaces for which the set of multipliers is a subset of the ball algebra.

Holomorphic functions on the quantum polydisk and on the quantum ball

We introduce and study noncommutative (or „quantized") versions of the algebras of holomorphic functions on the polydisk and on the ball in \mathbb C^n . Specifically, for each q\in\mathbb C\setminus\{ 0\} we construct Fréchet algebras \mathcal O_q(\mathbb D^n) and \mathcal O_q(\mathbb B^n) such that for q=1 they are isomorphic to the algebras of holomorphic functions on the open polydisk \mathbb D^n and on the open ball \mathbb B^n , respectively. In the case where 0 < q < 1 , we establish a relation between our holomorphic quantum ball algebra \mathcal O_q(\mathbb B^n) and L.L. Vaksman's algebra C_q(\bar{\mathbb B}^n) of continuous functions on the closed quantum ball. Finally, we show that \mathcal O_q(\mathbb D^n) and \mathcal O_q(\mathbb B^n) are not isomorphic provided that |q|=1 and n\ge 2 . This result can be interpreted as a q -analog of Poincaré's theorem, which asserts that \mathbb D^n and \mathbb B^n are not biholomorphically equivalent unless n=1 .

Bohr inequalities for free holomorphic functions on polyballs

Multivariable operator theory is used to provide Bohr inequalities for free holomorphic functions with operator coefficients on the regular polyball Bn, n=(n1,…,nk)∈Nk, which is a noncommutative analogue of the scalar polyball (Cn1)1×⋯×(Cnk)1. The Bohr radius Kmh(Bn) (resp. Kh(Bn)) associated with the multi-homogeneous (resp. homogeneous) power series expansions of the free holomorphic functions are the main objects of study in this paper. We extend a theorem of Bombieri and Bourgain for the disc D:={z∈C:|z|<1} to the polyball, and obtain the estimations 13k<Kmh(Bn)<2log⁡kk if k>1, extending Boas-Khavinson result for the scalar polydisc Dk.With respect to the homogeneous power series expansion, we prove that Kh(Bn)=1/3, extending the classical result, and obtain analogues of Carathéodory, Fejér, and Egerváry-Szász inequalities for free holomorphic functions with operator coefficients and positive real parts on the polyball. These results are used to provide multivariable analogues of Landau's inequality and Bohr's inequality when the norm is replaced by the numerical radius of an operator.When specialized to the regular polydisc Dk (which corresponds to the case n1=⋯=nk=1), we obtain new results concerning Bohr, Landau, Fejér, and Harnack inequalities for operator-valued holomorphic functions and k-pluriharmonic functions on the scalar polydisc Dk. The results of the paper can be used to obtain Bohr type inequalities for the noncommutative ball algebra An, the Hardy algebra Fn∞, and the C⁎-algebra C⁎(S), generated by the universal model S of the polyball Bn.

Cluster values for algebras of analytic functions

The Cluster Value Theorem is known for being a weak version of the classical Corona Theorem. Given a Banach space X, we study the Cluster Value Problem for the ball algebra Au(BX), the Banach algebra of all uniformly continuous holomorphic functions on the unit ball BX; and also for the Fréchet algebra Hb(X) of holomorphic functions of bounded type on X (more generally, for Hb(U), the algebra of holomorphic functions of bounded type on a given balanced open subset U⊂X). We show that Cluster Value Theorems hold for all of these algebras whenever the dual of X has the bounded approximation property. These results are an important advance in this problem, since the validity of these theorems was known only for trivial cases (where the spectrum is formed only by evaluation functionals) and for the infinite dimensional Hilbert space.As a consequence, we obtain weak analytic Nullstellensatz theorems and several structural results for the spectrum of these algebras.

Duality, convexity and peak interpolation in the Drury–Arveson space

We consider the closed algebra Ad generated by the polynomial multipliers on the Drury–Arveson space. We identify Ad⁎ as a direct sum of the preduals of the full multiplier algebra and of a commutative von Neumann algebra, and establish analogues of many classical results concerning the dual space of the ball algebra. These developments are deeply intertwined with the problem of peak interpolation for multipliers, and we generalize a theorem of Bishop–Carleson–Rudin to this setting by means of Choquet type integral representations. As a byproduct we shed some light on the nature of the extreme points of the unit ball of Ad⁎.

Perturbations of the ball algebra

Perturbations of the ball algebra

Noncoherence of some rings of holomorphic functions in several variables as an easy consequence of the one-variable case

Using the facts that the disk algebra and the Wiener algebra are not coherent, we prove that the polydisc algebra, the ball algebra, and the Wiener algebra of the polydisc are not coherent.

Some geometric properties of disk algebras

In this paper we study some geometrical properties of certain classes of uniform algebras, in particular the ball algebra Au(BX) of all uniformly continuous functions on the closed unit ball and holomorphic on the open unit ball of a complex Banach space X. We prove that Au(BX) has k-numerical index 1 for every k, the lushness and also the AHSP. Moreover, the disk algebra A(D), and more in general any uniform algebra whose Choquet boundary has no isolated points, is proved to have the polynomial Daugavet property. Most of those properties are extended to the vector valued version AX of a uniform algebra A.

The compactness of the sum of weighted composition operators on the ball algebra

In this paper, we investigate the compactness of the sum of weighted composition operators on the unit ball algebra, and give the characterization of compact differences of two weighted composition operators on the ball algebra. The connectness of the topological space consisting of non-zero weighted composition operators on the unit ball algebra is also studied. 2000 Mathematics Subject Classification. Primary: 47B33; Secondary: 47B38, 46E15, 32A36.

On unilateral weighted shifts in noncommutative operator theory

In this work, we investigate the condition for the functional model of unilateral weighted shift operator system. With the help of this model we also present the necessary and sufficient conditions for the algebra generated by this system to be isometrically isomorphic to the ball algebra.

On certain quotient modules of the Bergman module

We construct a class of quotient modules Q of the Bergman module over the ball algebra A(B n ) for which the Schatten-class membership ofthe commutators [Z* i , Z i ] of the module operators Z 1 ,..., Z n can be precisely prescribed.

Normal Hilbert modules over the ball algebra A(B)

The normal cohomology functor $Ext_ℵ$ is introduced from the category of all normal Hilbert modules over the ball algebra to the category of A(B)-modules. From the calculation of $Ext_ℵ$-groups, we show that every normal C(∂B)-extension of a normal Hilber

Singular parts of pluriharmonic measures

A measure μ defined on the complex sphere S is called pluriharmonic if its Poisson integral is a pluriharmonic function (in the unit ball of ℂn). A probability measure ρ is called representing if ∫Sfdp=f(0) for all f in the ball algebra. It is shown that singular parts of pluriharmonic measures and representing measures are mutually singular. Bibliography: 5 titles.

Representing measures for the disc algebra and for the ball algebra

We consider the set of representing measures at 0 for the disc and the ball algebra. The structure of the extreme elements of these sets is investigated. We give particular attention to representing measures for the 2-ball algebra which arise by lifting representing measures for the disc algebra. Introduction. Let X be a compact Hausdorff space, C(X) the algebra of continuous functions on X with the supremum norm, A a function algebra on X and Φ a multiplicative linear functional on A. The set of all probability measures on X which represent Φ is denoted by MΦ. As is well known, MΦ is a nonempty, weak∗-compact, convex subset of the set of all regular Borel measures, M(X), which is viewed as the dual space of C(X). For a compact X ⊂ C, the function algebra P (X) is defined as the closure of the holomorphic polynomials in C(X). Let B = B be the unit ball in C, n ≥ 2, S its boundary. Next, D is the open unit disc in C, D its closure and T its boundary. Traditionally P (D) is called the disc algebra and denoted by A(D) while P (S) is called the ball algebra and denoted by A(B). Note that by our definition a representing measure for A(D) will be defined on D, while a representing measure for A(B) will be defined on S. The motivation is that in this way we obtain an interesting set of representing measures for A(D), all of which can be “lifted”, to furnish representing measures on the sphere. From now on Φ will be point evaluation at a point a, usually 0, of the unit ball or disc and we will write MaS, respectively MaD, or when no confusion is possible Ma, instead of MΦ. 1991 Mathematics Subject Classification: 30H05, 32E25, 46J10, 46J15.

Homogeneous polynomials on the ball and polynomial bases

It is shown that the spaces of homogenous polynomials on the complex 2-ball are uniformly isomorphic tol∞-spaces. The argument is based on explicit constructions and the decomposition method. A new construction is given of bases in the spaceAN of monomials 1,z,z2, ...,zN−1 on the disc (due to Bochkarev [Boc]). Also using decomposition methods, the existence of a base in the ball algebra is obtained.

Closed Ideals in the Ball Algebra

Bulletin of the London Mathematical SocietyVolume 21, Issue 5 p. 469-474 Notes and papers Closed Ideals in the Ball Algebra Håkan Hedenmalm, Håkan Hedenmalm Department of Mathematics Uppsala University, Thunbergsvägen 3 S-75238 Uppsala SwedenSearch for more papers by this author Håkan Hedenmalm, Håkan Hedenmalm Department of Mathematics Uppsala University, Thunbergsvägen 3 S-75238 Uppsala SwedenSearch for more papers by this author First published: September 1989 https://doi.org/10.1112/blms/21.5.469Citations: 3AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volume21, Issue5September 1989Pages 469-474 RelatedInformation

Applications of the Spaces of Homogeneous Polynomials to some Problems on the Ball Algebra

Applications of the Spaces of Homogeneous Polynomials to some Problems on the Ball Algebra

Applications of the spaces of homogeneous polynomials to some problems on the ball algebra

Denote by B 2 {B_2} the unit ball in C 2 {{\mathbf {C}}^2} . The existence is shown of a uniformly bounded orthonormal basis in H 2 ( B 2 ) {H^2}({B_2}) , by constructing such systems in the spaces of homogeneous polynomials. In the second part of the paper, those spaces of homogeneous polynomials are exploited to disprove the existence of generalized analytic projections, the so-called ( i p − π p ) ({i_p} - {\pi _p}) property, for the ball algebra.

Projections and Isomorphisms of the Ball Algebra

Projections and Isomorphisms of the Ball Algebra

Inner functions in the unit ball of [formula omitted

Let B be the open unit ball of C n , n > 1. Let I (for “inner”) be the set of all u ϵ H ∘( B) that have ¦u¦ = 1 a.e. on the boundary S of B. Aleksandrov proved recently that there exist nonconstant u ϵ I. This paper strengthens his basic theorem and provides further information about I and the algebra Q generated by I. Let XY be the finite linear span of products xy, x ϵ X, y ϵ Y, and let ¦X¦ be the norm closure, in L ∞ = L ∞( S), of X. Some results: set I is dense in the unit ball of H ∞( B) in the compact-open topology. On S, Q ̄ Q is weak ∗-dense in L ∞, ¦ Q ̄ Q¦ does not contain H ∞, C(S) ⊂¦ Q ̄ H ∞¦ ≠ ¦ H ̄ ∞H ∞¦ ≠ L ∞ . (When n = 1, ¦Q¦ = H ∞ and ¦ Q ̄ Q¦ = L ∞ .) Every unimodular ƒ ϵ L ∞ is a pointwise limit a.e. of products u v ̄ , u ϵ I, ν ϵ I . The zeros of every ƒ ≢ 0 in the ball algebra (but not of every H ∞ -function) can be matched by those of some u ϵ I, as can any finite number of derivatives at 0 if ∥ƒ∥ ∞ < 1 . However, ƒ u cannot be bounded in B if u ϵ I is non-constant.