Invertibility in weak-star closed algebras of analytic functions

For K subset of C a compact subset and mu a positive finite Borel measure supported on K, let R-infinity(K, mu) be the weak-star closure in L-infinity(mu) of rational functions with poles off K. We show that if R-infinity(K, mu) has no non-trivial L-infinity summands and f is an element of R-infinity(K, mu), then f is invertible in R-infinity(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 o posed by J. Dudziak [6] in 1984.(c) 2023 Elsevier Inc. All rights reserved.