In Rudin's real and complex analysis section 9.22 there's an interesting application, at least to me, of banach algebras techniques. I am not a mathematician so I might be misunderstanding what this section is.
Essentially because $L^1$ is a Banach Algebra w.r.t. the convolution as multiplication operator the following equation is well defined
$$ \varphi(f * g) = \varphi(f)\varphi(g) \;\;\;\; (1) $$
After some analysis at the end of the section yields the following theorem
9.23 Theorem To every complex homomorphism $\varphi$ in $L^1$, except to $\varphi = 0$ there corresponds a unique $t \in \mathbb{R}$ such that $\varphi(f) = \hat{f}(t)$.
If I may dumb this down what it sounds to me is that the Fourier transform is defined by the $(1)$ uniquely.
Here's a question, are there any other transforms that can be defined in a similar way? I know that the fourier transform is generalized by the Gelfand-Transform, but this is not a tool I am extremely familiar with, yet.