12 citations · 21 across the 26 of their papers we have counts for
33 papers
A nontrivial uniform algebra regular on the Cantor set
J. F. Feinstein, Alexander J. Izzo
We prove the existence of a nontrivial uniform algebra that is logmodular and regular on the Cantor set. As a consequence, we obtain that for every compact metrizable space X witho…
Approximation by zero-free continuous maps
Alexander J. Izzo
We prove that if E a subset of an n-dimensional manifold, then every continuous R^n-valued map on E that is zero-free on the interior of E can be approximated in the fine topology,…
Weakly strongly regular uniform algebras
J. F. Feinstein, Alexander J. Izzo
Given a uniform algebra A on a compact Hausdorff space X and a point x in X, denote by M_x the ideal of functions in A that vanish at x and by J_x the ideal of functions in A that…
A nontrivial uniform algebra Dirichlet on its maximal ideal space
Alexander J. Izzo
It is shown that there exists a nontrivial uniform algebra that is Dirichlet on its maximal ideal space and has a dense set of elements that are exponentials. This answers a 65-yea…
A normal uniform algebra that fails to be strongly regular at a peak point
Alexander J. Izzo
It is shown that there exists a normal uniform algebra, on a compact metrizable space, that fails to be strongly regular at some peak point. This answers a 31-year-old question of…
Every connected bounded domain of holomorphy has connected boundary
Alexander J. Izzo
It is shown that every connected, bounded domain of holomorphy in , , has connected boundary.
A sharper Swiss cheese
Alexander J. Izzo
It is shown that there exists a compact planar set K such that the uniform algebra R(K) is nontrivial and strongly regular. This settles an issue raised by Donald Wilken 55 years a…
One-point Gleason parts and point derivations in uniform algebras
Swarup N. Ghosh, Alexander J. Izzo
It is shown that a uniform algebra can have a nonzero bounded point derivation while having no nontrivial Gleason parts. Conversely, a uniform algebra can have a nontrivial Gleason…
Polynomially convex sets whose union has nontrivial hull
Alexander J. Izzo
Several results concerning pairs of polynomially convex sets whose union is not even rationally convex are given. It is shown that there is no restriction on how two spaces can be…
A uniform algebra approach to an approximation theorem of Sahutoglu and Tikaradze
Timothy G. Clos, Alexander J. Izzo
Using methods from the theory of uniform algebras, we give a simple proof of an approximation result of Sahutoglu and Tikaradze with -pseudoconvex domains replaced by the…
Polynomial Hulls of Arcs and Curves II
Alexander J. Izzo
We prove that if a compact set E in complex Euclidean space is contained in an arc J, then there is a choice of J whose polynomial hull is the union of J and the polynomial hull of…
The set of bounded continuous nowhere locally uniformly continuous functions is not Borel
Alexander J. Izzo
It is known that for a nowhere locally compact metric space, the set of bounded continuous, nowhere locally uniformly continuous real-valued functions on contains a dense $…
Polynomially Convex Arcs in Polynomially Convex Simple Closed Curves
Alexander J. Izzo, Edgar Lee Stout
We prove that every polynomially convex arc is contained in a polynomially convex simple closed curve. We also establish results about polynomial hulls of arcs and curves that are…
The convergence of hulls of curves
Alexander J. Izzo, Edgar Lee Stout
It is shown that a simple closed curve in that is a uniform limit of rectifiable simple closed curves each of which has nontrivial polynomial hull has itself nontrivi…
Weak Sequential Completeness of Uniform Algebras
J. F. Feinstein, Alexander J. Izzo
We give a simple, elementary proof that a uniform algebra is weakly sequentially complete if and only if it is finite-dimensional.
Polynomial Hulls of Arcs and Curves
Alexander J. Izzo
It is shown that there exist arcs and simple closed curves in with nontrivial polynomial hulls that contain no analytic discs. It is also shown that in any bounded…
Gleason parts and point derivations for uniform algebras with dense invertible group II
Alexander J. Izzo
Due to the omission of a hypothesis from an elementary lemma in the author's paper "Gleason parts and point derivations for uniform algebras with dense invertible group", some of t…
A doubly generated uniform algebra with a one-point Gleason part off its Shilov boundary
Alexander J. Izzo
It is shown that there exists a compact set in with a nontrivial polynomial hull such that some point of is a one-point Gl…
Topology of Gleason Parts in maximal ideal spaces with no analytic discs
Alexander J. Izzo, Dimitris Papathanasiou
We strengthen, in various directions, the theorem of Garnett that every -compact, completely regular space occurs as a Gleason part for some uniform algebra. In particular,…
A hull that contains no discs
Alexander J. Izzo
It is shown that if is a unital commutative Banach algebra with a dense set of invertible elements, then the maximal ideal space of contains no compact, locally connected,…
No topological condition implies equality of polynomial and rational hulls
Alexander J. Izzo
It is shown that no purely topological condition implies the equality of the polynomial and rational hulls of a set: For any compact subset of a Euclidean space, there exists a…
A Cantor set whose polynomial hull contains no analytic discs
Alexander J. Izzo, Norman Levenberg
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…
Spaces with polynomial hulls that contain no analytic discs
Alexander J. Izzo
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 conve…
Analytic Discs and Uniform Algebras on Real-Analytic Varieties
Alexander J. Izzo
Under very general conditions it is shown that if is a uniform algebra generated by real-analytic functions, then either consists of all continuous functions or else there…