Weakly Peripherally-Multiplicative Operators Between Uniform Algebras

Document Type

Presentation Abstract

Presentation Date

12-7-2006

Abstract

Let X be a compact Hausdorff space. A uniform algebra is a sub-algebra of continuous functions, ƒ : X→ ℂ with the uniform norm. In this case the spectrum is the range i.e. σ(ƒ)=ƒ(X) and σg(ƒ):{λ ∈σ(ƒ):∣λ∣ = ∥ƒ∥. The mere existence of this norm in the definition places restrictions on the algebraic structure so that a mapping between uniform algebras that preserves certain analytic conditions is necessarily an algebraic isomorphism. There are many results with this theme. For example Aaron Luttman and Thomas Tonev have produced the following:

If T:A→B is a surjective, unital mapping between uniform algebras that is peripherally multiplicative (i.e. for all ƒ, g∊A,σg(ƒg)=σg (TƒTg)) the T is an algebraic isomorphism.

In this talk we prove a stronger result: If T:A→B is a mapping between uniform algebras that preserves the peaking functions and is weakly-peripherally multiplicative (σg(ƒg)∩σg(TƒTg)≠Ø for all ƒ, g∊A) the T is an algebraic isomorphism. (A function h∊A is a peaking function if σg(h)={1})

Also some extensions of this theorem will be discussed.

Additional Details

Thursday, 30 November 2006
4:10 p.m. in Math 109

This document is currently not available here.

Share

COinS