Let f:A\(\to B\) be a homomorphism of (complex unital commutative) Banach algebras. Let \(f^*:X(B)\to X(A)\) be the transpose of f: for every character \(h:B\to {\mathbb{C}},f^*(h):=h.f\). The main result says that \(f^*\) is surjective if and only if for every \(n\in {\mathbb{N}}\) \(f^{- 1}(U_ n(B))=U_ n(A),\) where for a Banach algebra D we set \(U_ n(D):=\{d\in D^ n:\sum^{n}_{i=1}Dd_ i=D\}.\)
Several corollaries are obtained concerning: a) the spectrum of \(H^{\infty}\); b) the cortex of a Banach algebra; c) the extension of characters of a Banach algebra A to the superalgebras of A.
Concerning this type of results see also ”Extensions of characters and generalized Shilov boundaries” by G. Corach and A. Maestripieri, Rev. Union Mat. Argentina 32, 211-216 (1986) and ”Generalized rational convexity in Banach algebras” by G. Corach and F. D. Suárez, Trabajos de Matemática, IAM, \(N\circ 114\) (1987), (preprint).