# Noncommutative Uniform Algebras

## Document Type

Presentation Abstract

## Presentation Date

5-16-2008

## Abstract

A uniform algebra A is a Banach algebra such that ||*f*^{2}||= ||f||^{2}, for all f ∈ A. It is well known that any *complex* uniform Banach algebra is automatically commutative and is isometrically isomorphic with a subalgebra of C_{C}(X) [Hirschfeld and Żelazko, 1968]; such algebras are the most classical and well studied ones. Commutative, real uniform algebras have also been studied for years. Any such algebra *A* is isometrically isomorphic with a real subalgebra of C_{C}(X) for some compact set *X*; furthermore in most cases *X* can be just divided into three parts X_{1}, X_{2} and X_{3} such that A_{|x1} is a complex uniform algebra, A_{|x2} consists of complex conjugates of the functions from A_{|x1}, and A_{|x3} is equal to C_{H}(*X*_{3}). In case of real uniform algebras the condition ||*f*^{2}||= ||f||^{2} no longer implies commutativity - the algebra of quaternions serves as the simples counterexample. On the other hand there have been very little study of non commutative real uniform algebras. We show that any such algebra is isometrically isomorphic with a real subalgebra of C_{H}(*X*) - the algebra of all continuous functions defined on a compact set *X* and taking values in the field **H** of quaternions. We will also produce a general structural description of such algebras and address the question whether there are non trivial example of such algebras other then the direct sum of the entire algebra C_{H}(*X*) and a commutative real uniform algebra.

## Recommended Citation

Jarosz, Krzysztof, "Noncommutative Uniform Algebras" (2008). *Colloquia of the Department of Mathematical Sciences*. 299.

https://scholarworks.umt.edu/mathcolloquia/299

## Additional Details

Friday, 16 May 2008

2:10 p.m. in 103

1:30 p.m. Refreshments in Math Lounge 109