Invertibility in Weak-Star Closed Algebras of Analytic Functions

For $K\subset \mathbb{C}$ a compact subset and $\mu$ a positive finite Bore1 measure supported on $K,$ let $R^{\mathrm{\infty }}(K,\mu )$ be the weak-star closure in $L^{\mathrm{\infty }}(\mu )$ of rational functions with poles off $K\mathrm{.}$ We show that if $R^{\mathrm{\infty }}(K,\mu )$ has no non-trivial $L^{\mathrm{\infty }}$ summands and $f\in R^{\mathrm{\infty }}(K,\mu ),$ then $f$ is invertible in $R^{\mathrm{\infty }}(K,\mu )$ 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 $\mu \mathrm{.}$ The result proves the conjecture $\diamond$ posed by J. Dudziak in 1984.