A solution to the Bézout equation in A(K) without Gelfand theory

摘要

Let M ? ? be compact and K ? M closed, and let A(K, M) be the uniform algebra of all functions continuous on M and holomorphic in the interior K° of K. We present a constructive proof of Arens' classical result that for (f _1, ..., f _n) ∈ A(K, M) ~n the Bézout equation has a solution in A(K, M) if and only if the functions f _j have no common zero in K. We shall also consider matrix-valued Bézout equations.