# Extensions of uniform algebras

## TL;DR

This paper explores new, more general classes of uniform algebra extensions, expanding on existing methods like Cole's, and demonstrates that key properties are preserved in these broader extensions.

## Contribution

Introduces a new class of uniform algebra extensions and shows they retain important properties, broadening the scope of algebraic extension techniques.

## Key findings

- Several important properties are preserved in the new extensions.
- Well-known examples are shown to belong to these new classes.
- The new classes generalize existing extension methods.

## Abstract

The theory of algebraic extensions of Banach algebras is well established, and there are many constructions which yield interesting extensions. In particular, Cole's method for extending uniform algebras by adding square roots of functions to a given uniform algebra has been used to solve many problems within uniform algebra theory. However, there are numerous other examples in the theory of uniform algebras that can be realised as extensions of a uniform algebra, and these more general extensions have received little attention in the literature. In this paper, we investigate more general classes of uniform algebra extensions. We introduce a new class of extensions of uniform algebras, and show that several important properties of the original uniform algebra are preserved in these extensions. We also show that several well-known examples from the theory of uniform algebras belong to these more general classes of uniform algebra extensions.