Gleason parts of bidual algebras

TL;DR

This paper proves that Gleason parts of a uniform algebra embed into the spectrum of its second dual as entire Gleason parts, using the equality of weak-star and norm topologies on Bear-Gleason parts.

Contribution

It establishes that the embedding of Gleason parts into the spectrum of the second dual preserves the Gleason part structure, based on topology equality results.

Findings

Embedding of Gleason parts into the second dual spectrum is an entire Gleason part.

Weak-star and norm topologies coincide on Bear-Gleason parts.

The result applies to all Gleason parts of uniform algebras.

Abstract

It is shown that the embeding of any Gleason part of a uniform algebra into the spectrum of its second dual is an entire Gleason part. This result is based on the equality of weak-star and norm topologies on the Bear-Gleason part.