Countable linear combinations of characters on commutative Banach algebras

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 $ϕ_n$ on the disc algebra and an $\ell_1$ sequence of complex numbers $λ_n$, not all zero, such that $\sum_{n=1}^\infty λ_n ϕ_n =0.$ 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.

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