# Topology of Gleason Parts in maximal ideal spaces with no analytic discs

## TL;DR

This paper extends Garnett's theorem by constructing uniform algebras with maximal ideal spaces that contain prescribed Gleason parts without analytic discs, including metrizable and Euclidean subspace cases.

## Contribution

It demonstrates the existence of uniform algebras with maximal ideal spaces containing specific Gleason parts and no analytic discs, strengthening previous results.

## Key findings

- Maximal ideal spaces can be constructed with prescribed Gleason parts.
- Such spaces can be made metrizable when the original space is metrizable.
- Existence of compact sets in complex Euclidean spaces with Gleason parts homeomorphic to given spaces.

## Abstract

We strengthen, in various directions, the theorem of Garnett that every $\sigma$-compact, completely regular space $X$ occurs as a Gleason part for some uniform algebra. In particular, we show that the uniform algebra can always be chosen so that its maximal ideal space contains no analytic discs. We show that when the space $X$ is metrizable, the uniform algebra can be chosen so that its maximal ideal space is metrizable as well. We also show that for every locally compact subspace $X$ of a Euclidean space, there is a compact set $K$ in some ${\mathbb C}^N$ so that $\hat K \setminus K$ contains a Gleason part homeomorphic to $X$ and $\hat K$ contains no analytic discs.