TITLE: Where does the Laplace transform come from?
QUESTION [7 upvotes]: 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?

REPLY [21 votes]: 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.