http://mathworld.wolfram.com/GelfandTransform.html
In the definition, what does $x$, $\hat x(\phi)$, and $\phi$ represent exactly if we were to consider definition of the Fourier transform? Can someone please help make the connection between Gelfand transform and Fourier transform more explicit?
Where Fourier transform is taken to be the usual operator: $$ \mathcal{F}(\omega) = \int\limits_{-\infty}^{\infty} \mathcal{f}(t) \, e^{-i\omega x} \, \mathrm{d} t $$
If $\phi$ represents the transform itself, I cannot see how it would satisfy $\phi(xy) = \phi(x)\phi(y)$