A proof of the ordinary Gleason-Kahane-Żelazko theorem for complex functionals

The Theorem(Gleason-Kahane-Żelazko) If $\phi$ is a complex linear functional on a unitary Banach algebra $A$, such that $\phi(e)=1$ and $\phi(x) \neq 0$ for every invertible $x \in A$, then\phi(xy)=\phi(x)\phi(y)Namely, $\phi$ is a complex homomorphism. Notations and remarksSuppose $A$ is a complex unitary Banach algebra and $\phi: A \to \mathbb{C}$ is a linear functional which is not identically $0$ (for convenience), and if \phi(xy)=\phi(x)\phi(y)for all $x \in A$ and $y \in A$, then $\phi$ is...

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