Mathematics Ph.D. Dissertations

Isolated Point Theorems for Uniform Algebras on Manifolds

Date of Award

2014

Document Type

Dissertation

Degree Name

Doctor of Philosophy (Ph.D.)

Department

Mathematics/Mathematics (Pure)

First Advisor

Alexander Izzo (Advisor)

Second Advisor

Lewis Fulcher (Committee Member)

Third Advisor

Juan Bes (Committee Member)

Fourth Advisor

Kit Chan (Committee Member)

Fifth Advisor

Steven Seubert (Committee Member)

Abstract

Suppose A is a uniform algebra on a compact Hausdorff space X. In 1957, Andrew Gleason conjectured that if (i) the maximal ideal space of A is X, and (ii) each point of X is a one-point Gleason part for A, then A must be C(X), the collection of all complex-valued continuous functions on X. Subsequently, a weaker conjecture, known as Peak Point Conjecture, was considered in which condition (ii) was replaced by the stronger condition that "each point of X is a peak point for A". In fact, one can consider a stronger conjecture, referred as Isolated Point Conjecture, by considering a weaker condition "each point of X is isolated in the Gleason metric for A" in place of condition (ii). However, all of these three conjectures fail by a counterexample produced by Brian Cole in 1968. In 2001, John Anderson and Alexander Izzo proved that the Peak Point Conjecture is true for uniform algebras generated by collections of C1 functions on a compact two-dimensional real manifold-with-boundary of class C1. In the same year, Anderson, Izzo and John Wermer together proved that the same conjecture is true for uniform algebras generated by polynomials on compact subsets of real-analytic three-dimensional submanifolds of complex Euclidean spaces. In this dissertation, we will prove Gleason's conjecture, and the Isolated Point Conjecture for the earlier mentioned classes of uniform algebras considered by Anderson, Izzo and Wermer. In view of the relations of isolated point (in the Gleason metric) with Gleason part and peak point, it is sufficient to consider the Isolated Point Conjecture, the strongest of all the three conjectures. More explicitly, we will prove that the Isolated Point Conjecture is true for uniform algebras generated by collections of C1 functions on a compact two-dimensional real manifold-with-boundary of class C1, as well as for uniform algebras generated by polynomials on compact subsets of real-analytic three-dimensional submanifolds of complex Euclidean spaces. Hence, in particular, these results will generalize the corresponding results proved by Anderson, Izzo and Wermer.

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