The Gelfand transform on the commutative Banach \*-algebra $L^1(\mathbb{R})$ is just the Fourier transform.


Q. What can we say concerning the Laplace transform?



Where does the Laplace transform come from?
Set $A = L^1([0, \infty))$, equipped with the structure of a Banach $\*$-algebra via convolution. The spectrum of this algebra is the half plane $\text{Re}(z) \geq 0$, and the Gelfand transform is the Laplace transform.


