Topology of Gleason Parts in Maximal Ideal Spaces with no Analytic Discs

Uniform Approximation and Maximal Ideal Spaces

Let X be a compact set in the z-plane. We are interested in two function spaces associated with X: C(X) — space of all continuous complex-valued functions on X. P(X) =space of all uniform limits of polynomials on X. Thus a function ƒ on X lies in P{X) if there exists a sequence {Pn} of polynomials converging to ƒ uniformly on X. Clearly P(X) is part of C(X). QUESTION I. When is P(X) = C(X)t i.e...

Topology of the Maximal Ideal Space of H

We study the structure of the maximal ideal space M(H∞) of the algebra H ∞ = H∞(D) of bounded analytic functions defined on the open unit disk D ⊂ C. Based on the fact that dim M(H∞) = 2 we prove for H∞ the matrix-valued corona theorem. Our results heavily rely on the topological construction describing maximal ideal spaces of certain algebras of continuous functions defined on the covering spa...

Certain subalgebras of Lipschitz algebras of infinitely differentiable functions and their maximal ideal spaces

We study an interesting class of Banach function algebras of innitely dierentiable functions onperfect, compact plane sets. These algebras were introduced by Honary and Mahyar in 1999, calledLipschitz algebras of innitely dierentiable functions and denoted by Lip(X;M; ), where X is aperfect, compact plane set, M = fMng1n=0 is a sequence of positive numbers such that M0 = 1 and(m+n)!Mm+n ( m!Mm)...

Topology Proceedings SPACES WITH NO INFINITE DISCRETE SUBSPACE

We show that the spaces with no infinite discrete subspace are exactly those in which every closed set is a finite union of irreducibles. Call them FAC spaces: this generalizes a theorem by Erdős and Tarski (1943), according to which a preordered set has no infinite antichain—the finite antichain, or FAC, property— if and only if all its downwards-closed subsets are finite unions of ideals. All...

A Geometric Characterization of Gleason Parts