Invertibility in weak-star closed algebras of analytic functions

Abstract

For K⊂C a compact subset and μ a positive finite Borel measure supported on K, let R∞(K,μ) be the weak-star closure in L∞(μ) of rational functions with poles off K. We show that if R∞(K,μ) has no non-trivial L∞ summands and f∈R∞(K,μ), then f is invertible in R∞(K,μ) if and only if Chaumat's map for K and μ applied to f is bounded away from zero on the envelope with respect to K and μ. The result proves the conjecture ⋄ posed by J. Dudziak [6] in 1984.