Spaces with polynomial hulls that contain no analytic discs

Abstract

Extensions of the notions of polynomial and rational hull are introduced. Using these notions, a generalization of a result of Duval and Levenberg on polynomial hulls containing no analytic discs is presented. As a consequence it is shown that there exists a Cantor set in C 3 with a nontrivial polynomial hull that contains no analytic discs. Using this Cantor set, it is shown that there exist arcs and simple closed curves in C 4 with nontrivial polynomial hulls that contain no analytic discs. This answers a question raised by Bercovici in 2014 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 C N , for some N , in such a way as to have a nontrivial polynomial 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.

A doubly generated uniform algebra with a one-point Gleason part off its Shilov boundary

“…Using results from the author's paper [8], we will obtain the following as another corollary. Corollory 1.4.…”

“…The first of these is standard, and a short proof can be found in [12,Lemma 1.7.4]. The others are proven in [8].…”

“…The proof of Theorem 1.1 has some similarities with the construction of a polynomial hull without analytic discs given by Julien Duval and Norman Levenberg [5] (or see [12,Lemma 1.7.5]). The proof of Theorem 1.2 involves combining the ingredients in the proof of Theorem 1.1 with ingredients in the proof of [8,Theorem 5.2]. (Theorem 5.2 of [8] generalizes [7,Theorem 4.1] whose proof was inspired by the constructions of Duval and Levenberg [5] and of Dales and Feinstein [4].…”

“…[8, Lemma 4.1] Let K be a collection of compact sets in C N totally ordered by inclusion. Let K ∞ = K∈K K. Then K k ∞ = K∈K K k and h k r (K ∞ ) = K∈K h k r (K).…”

Polynomial hulls of arcs and curves

“…Consequently, by [13, Corollary 1.6.8] for instance, P ( J) = {f ∈ C( J) : f | E ∈ P ( E)}. Therefore, the density of invertible elements in P (E) implies that P (J) has dense invertibles by [7,Lemma 8.1]. Furthermore, J \ J = E \ J and the set E \ J must be nonempty as the two-dimensional Hausdorff measure of E \ E can not be zero, by [13, Corollary 1.6.8] for instance (and in fact must be infinite by [2,Theorem 21.9]).…”