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$L^1(\mathbb R^n)$, $L^1(\mathbb R_+)$, $C^0_c(\mathbb R_+)$, $C^\infty_c(\mathbb R_+)$ are algebras of convolution.

Question 1: is there a classification of subalgebras of convolution of $L^1(\mathbb R^n)$?

Question 2: is there a classification of subalgebras of convolution of $\mathscr E'(\mathbb R^n)$ (distributions on $\mathbb R^n $ with compact support)?

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  • $\begingroup$ Just a thought: I would start by looking at Fourier transforms, so that convolution is turned into multiplication. Then Stone–Weierstrass suggests that subalgebras could be classified by conditions of the form: $\hat{f}(z)$ is constant on a fixed collection of (pairwise disjoint) subsets $A_\nu$ or $\mathbb{R} \cup \{\infty\}$. $\endgroup$
    – Mateusz Kwaśnicki
    Commented May 6, 2017 at 23:06

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