Countable linear combinations of characters on commutative Banach algebras

TL;DR

The paper explores conditions under which countable linear combinations of characters on commutative Banach algebras can sum to zero, extending Wolff's 1921 result and identifying topological constraints for such combinations.

Contribution

It generalizes Wolff's classical result to broader classes of commutative Banach algebras and establishes topological conditions preventing such linear combinations.

Findings

Wolff's result applies to the disc algebra with specific sequences.

In general commutative Banach algebras, such linear combinations are impossible if the character set's closure lacks perfect subsets.

The paper identifies topological conditions that restrict the existence of non-trivial linear combinations of characters.

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.