Invertibility in Weak-Star Closed Algebras of Analytic Functions

Abstract

For $K\subset \mathbb{C}$ a compact subset and $\hat{I}¼$ a positive finite Bore1 measure supported on $K,$ let $R^{\infty }(K,\hat{I}¼)$ be the weak-star closure in $L^{\infty }(\hat{I}¼)$ of rational functions with poles off $K.$ We show that if $R^{\infty }(K,\hat{I}¼)$ has no non-trivial $L^{\infty }$ summands and $f\in R^{\infty }(K,\hat{I}¼),$ then $f$ is invertible in $R^{\infty }(K,\hat{I}¼)$ if and only if Chaumat's map for $K$ and $\hat{I}¼$ applied to $f$ is bounded away from zero on the envelope with respect to $K$ and $\hat{I}¼.$ The result proves the conjecture $\diamond$ posed by J. Dudziak in 1984.

J. Funct. Anal. (2023). arXiv admin note: text overlap with arXiv:2212.10811