Countable linear combinations of characters on commutative Banach algebras

Abstract

An elegant but elementary result of Wolff from 1921, when interpreted in terms of Banach algebras, shows that it is possible to find a sequence of distinct characters $\phi_n$ on the disc algebra and an $\ell_1$ sequence of complex numbers $\lambda_n$, not all zero, such that $\sum_{n=1}^\infty \lambda_n \phi_n =0.$ We observe that, even for general commutative, unital Banach algebras, this is not possible if the closure of the countable set of characters has no perfect subsets.