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Reading about Gelfand-Naimark theorem I've seen that the Fourier transform is a special case of Gelfand transform for the space $L^1(\mathbb{R})$ with the convolution product. In a related question on this site (Fourier transform as a Gelfand transform) I see (if I well understand) that this is a case of Pontryagin duality. Now my question is: also other transforms, as Laplace transform, are cases of Gelfand transform? And, if yes, what is the suitable algebra of functions in which we can see such correspondence?

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If $L^1(\mathbb{R}^+)$ is equipped with the convolution product $$(f \star g)(t) = \int_0^tf(t − s)\cdot g(s) \,ds\qquad, \forall f, g \in L^1(\mathbb{R}^+)$$ then it becomes a Banach algebra. The Gelfand theory here corresponds to the Laplace transform.

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  • $\begingroup$ Thanks. You know if this result can be generalized to other transforms? As Hankel transform that use Bessel functions as a basis. And to functions on $\mathbb{R}^n$ ? $\endgroup$ – Emilio Novati Feb 25 '15 at 9:46
  • $\begingroup$ I would recommend that you look at a good book on Functional Analysis to learn more about the Gelfand Transform. $\endgroup$ – JP McCarthy Feb 25 '15 at 10:35

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