Invertibility in Weak-Star Closed Algebras of Analytic Functions

TL;DR

This paper characterizes invertibility in certain weak-star closed algebras of analytic functions, proving a conjecture by J. Dudziak that relates invertibility to boundedness conditions on a specific map.

Contribution

It establishes a necessary and sufficient condition for invertibility in $R^ ^ ext{infty}(K,\mu)$, confirming Dudziak's 1984 conjecture.

Findings

Invertibility characterized by Chaumat's map bounded away from zero.

Proves Dudziak's conjecture from 1984.

Provides conditions for invertibility in weak-star closed algebras.

Abstract

For $K\subset \mathbb C$ a compact subset and $\mu$ a positive finite Bore1 measure supported on $K,$ let $R^\infty (K,\mu)$ be the weak-star closure in $L^\infty (\mu)$ of rational functions with poles off $K.$ We show that if $R^\infty (K,\mu)$ has no non-trivial $L^\infty$ summands and $f\in R^\infty (K,\mu),$ then $f$ is invertible in $R^\infty (K,\mu)$ if and only if Chaumat's map for $K$ and $\mu$ applied to $f$ is bounded away from zero on the envelope with respect to $K$ and $\mu.$ The result proves the conjecture $\diamond$ posed by J. Dudziak in 1984.