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Alexander J. Izzo

Publications and source records attributed to Alexander J. Izzo.

33 records · Page 2Linked to original sources
arXiv2019

No topological condition implies equality of polynomial and rational hulls

It is shown that no purely topological condition implies the equality of the polynomial and rational hulls of a set: For any compact subset $K$ of a Euclidean space, there exists a set $X$, in some ${\mathbb C}^N$, that is homeomorphic to $K$ and is rationally convex but not polynomially convex. In addition, it is shown that for the surfaces in ${\mathbb C}^3$ constructed by Izzo and Stout, whose polynomial hulls are nontrivial but contain no analytic discs, the polynomial and rational hulls coincide, thereby answering a question of Gupta. Equality of polynomial and rational hulls is shown also for $m$-dimensional manifolds ($m\geq 2$) with polynomial hulls containing no analytic discs constructed by Izzo, Samuelsson Kalm, and Wold and by Arosio and Wold.

math.CV↗
arXiv2019

A Cantor set whose polynomial hull contains no analytic discs

A generalization of a result of Wermer concerning the existence of polynomial hulls without analytic discs is presented. As a consequence it is shown that there exists a Cantor set $X$ in ${\mathbb C}^3$ whose polynomial hull is strictly larger than $X$ but contains no analytic discs.

math.CV↗
arXiv2019

Spaces with polynomial hulls that contain no analytic discs

Extensions of the notions of polynomially and rationally convex hulls are introduced. Using these notions, a generalization of a result of Duval and Levenberg on polynomially convex hulls containing no analytic discs is presented. As a consequence it is shown that there exists a Cantor set $X$ in ${\mathbb C}^3$ with a nontrivial polynomially convex hull that contains no analytic discs. Using this Cantor set, it is shown that there exist arcs and curves in ${\mathbb C}^4$ with nontrivial polynomially convex hulls that contain no analytic discs. This answers a question raised a few years ago by Bercovici and can be regarded as a partial answer to a question raised by Wermer over 60 years ago. More generally, it is shown that every uncountable, compact subspace of a Euclidean space can be embedded as a subspace $X$ of ${\mathbb C}^N$, for some N, in such a way as to have a nontrivial polynomially convex hull that contains no analytic discs. In the case when the topological dimension of the space is at most one, $X$ can be chosen so as to have the stronger property that $P(X)$ has a dense set of invertible elements.

math.CV↗
arXiv2018

A hull that contains no discs

It is shown that if $A$ is a unital commutative Banach algebra with a dense set of invertible elements, then the maximal ideal space of $A$ contains no compact, locally connected, simply coconnected subspace of topological dimension $\geq 2$. As a consequence, the existence of a compact set in ${\mathbb C}^2$ with a nontrivial polynomial hull that contains no topological discs is obtained. This strengthens the celebrated result of Stolzenberg from 1963 that there exists a nontrivial polynomial hull that contains no analytic discs, and it answers a question stated in the literature 10 years ago by Dales and Feinstein but considered much earlier.

math.CV↗
arXiv2018

A General Method for Constructing Essential Uniform Algebras

A general method for constructing essential uniform algebras with prescribed properties is presented. Using the method, the following examples are constructed: an essential, natural, regular uniform algebra on the closed unit disc; an essential, natural counterexample to the peak point conjecture on each manifold of dimension at least three; and an essential, natural uniform algebra on the unit sphere in C^3 containing the ball algebra and invariant under the action of the 3-torus. These examples show that a smoothness hypothesis in some results of Anderson and Izzo cannot be omitted.

math.FA↗
arXiv2017

Analytic Discs and Uniform Algebras on Real-Analytic Varieties

Under very general conditions it is shown that if $A$ is a uniform algebra generated by real-analytic functions, then either $A$ consists of all continuous functions or else there exists a disc on which every function in $A$ is holomorphic. This strengthens several earlier results concerning uniform algebras generated by real-analytic functions.

math.CV↗
arXiv2016

Hulls of Surfaces

In this paper it is shown that every compact two-dimensional manifold $S$, with or without boundary, can be embedded in $\mathbb C^3$ as a smooth submanifold $Σ$ in such a way that the polynomially convex hull of $Σ$, though strictly larger than $Σ$, contains no analytic disc.

math.CV↗
arXiv2016

Localization for Uniform Algebras Generated by Real-Analytic Functions

It is shown that if $A$ is a uniform algebra generated by real-analytic functions on a suitable compact subset $K$ of a real-analytic variety such that the maximal ideal space of $A$ is $K$, and every continuous function on $K$ is locally a uniform limit of functions in $A$, then $A=C(K)$. This gives an affirmative answer to a special case of a question from the Proceedings of the Symposium on Function Algebras held at Tulane University in 1965.

math.CV↗
arXiv2016

Pick and Peak Interpolation

We show how Pick interpolation and interpolation on peak interpolation sets can be combined in an abstract uniform algebra setting. In particular as a special case, the Rudin-Carleson theorem can be combined with the classical Pick interpolation theorem on the disc.

math.CV↗
arXiv2016

Gleason parts and point derivations for uniform algebras with dense invertible group

It is shown that there exists a compact set $X$ in ${\bf C}^N$ ($N\geq 2$) such that $\widehat X\setminus X$ is nonempty and the uniform algebra $P(X)$ has a dense set of invertible elements, a large Gleason part, and an abundance of nonzero bounded point derivations. The existence of a Swiss cheese $X$ such that $R(X)$ has a Gleason part of full planar measure and a nonzero bounded point derivation at almost every point is established. An analogous result in ${\bf C}^N$ is presented. The analogue for rational hulls of a result of Duval and Levenberg on polynomial hulls containing no analytic discs is established. The results presented address questions raised by Dales and Feinstein.

math.CV↗
arXiv2016

A Hull with No Nontrivial Gleason Parts

The existence of a nontrivial polynomially convex hull with every point a one-point Gleason part and with no nonzero bounded point derivations is established. This strengthens the \hbox{celebrated} result of Stolzenberg that there exists a nontrivial polynomially convex hull that contains no analytic discs. A doubly generated counterexample to the peak point conjecture is also presented.

math.CV↗
arXiv2015

A Peak Point Theorem for Uniform Algebras on Real-Analytic Varieties

It was once conjectured that if $A$ is a uniform algebra on its maximal ideal space $X$, and if each point of $X$ is a peak point for $A$, then $A = C(X)$. This peak-point conjecture was disproved by Brian Cole in 1968. Here we establish a peak-point theorem for uniform algebras generated by real-analytic functions on real-analytic varieties, generalizing previous results of the authors and John Wermer.

math.CV↗
arXiv2014

Presence or absence of analytic structure in maximal ideal spaces

We study extensions of Wermer's maximality theorem to several complex variables. We exhibit various smoothly embedded manifolds in complex Euclidean space whose hulls are non-trivial but contain no analytic disks. We answer a question posed by Lee Stout concerning the existence of analytic structure for a uniform algebra whose maximal ideal space is a manifold.

math.CV↗