# Fourier transform as a Gelfand transform

One question came to my mind while looking at the proof of Gelfand-Naimark theorem. Is Fourier transform a kind of Gelfand transform? Are there any other well-known transforms which are so?

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any.. i need some real examples of gefland transform – Koushik Dec 31 '12 at 11:17
A good example is $L^1(\Bbb R)$ where the product is convolution. We need to add an unity element, hence we work on $L^1\times\Bbb R$ with the operation $(f_1,r_1)+(f_2,r_2)=(f_1+f_2,r_1+r_2)$ and $(f_1,r_1)*(f_2,r_2)=(f_1*f_2+r_1f_2+r_2f_1,r_1r_2)$. We can give a characterization of the homomorphisms: there are $$h_t(f,r):=\int_{\Bbb R}e^{its}f(s)ds$$ and $h_{\infty}(f,r)=r$. – Davide Giraudo Dec 31 '12 at 11:20

Yes, the Fourier transform is the Gelfand transform for the commutative Banach algebras $(L_1(G),*)$ where $G$ is a locally compact commutative group. One can show that there is a bijection between the homomorphisms from $(L_1(G),*)$ into $\mathbb{C}$ and the elements of the Pontryagin dual $\hat{G}$, which is used to define the Fourier transform.