Correspondence between maximal ideals and multiplicative functionals of a non unital, commutative Banach algebra.

The kernels of nonzero homomorphisms to $\mathbb C$ are modular ideals, terminology that might help you find more references.

Without any further restriction on the algebras, using the zero product is a way to provide trivial counterexamples. E.g., take $\mathbb C$ with the $0$ product, which has maximal ideal $\{0\}$ and no nonzero homomorphisms to $\mathbb C$.

Googling led me to the following maybe more interesting example, Example 1.3 in this Feinstein and Somerset article: Take $C[0,1]$ with its usual Banach space structure but with multiplication $(f\diamond g)(t) = f(t)g(t)t$. Then the ideal of functions vanishing at $0$ is maximal but not the kernel of a nonzero homomorphism to $\mathbb C$.

Linear-multiplicative functionals (aka characters) on complex Banach algebras are automatically continuous, so their kernels are closed. (You will find a slick proof of this fact on p. 181 of Allan's and Dales' Introduction to Banach Spaces and Algebras.) However, in the non-unital case it may well happen that a maximal ideal is dense.

The right notion to look at is the notion of a maximal modular ideal. Such ideals are bijectively associated to (kernels of) characters.

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