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Let $L$ be the Banach algebra of $L^1$-functions from $\mathbb{R}$ to $\mathbb{C}$ with $L^1$-norm and convolution as algebra multiplication. Assume that we realized knew that all the homomorphisms from $L$ to $\mathbb{C}$ are identical the zero map and evaluation of the Fourier transform values in specific at individual real pointsnumbers: $f\rightarrow f \int mapsto \int_{\mathbb R} f(t)e^{it\alpha}dt$ . for some real $\alpha$. We may add unity a unit $e$ to $L$ artifically, just artificially by considering the new Banach algebra $A:=L\oplus \mathbb{C}\cdot e$ with natural operations. Then the fact that any $L^1$-function with zero whose Fourier transform is zero must be zero itself may be rephrased algebraically: the algebra $A$ is semisimple (as maximal ideals of unital banach Banach algebras correspond to homomorphisms to $\mathbb{C}$ by the Gelfand-Mazur theorem).

My question is whether this may be proved a priori and independently (and maybe for some wide class of commutative unital Banach algebras).

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# uniqueness theorem of Fourier transform: is there algebraic proof?

Let $L$ be the Banach algebra of $L^1$-functions from $\mathbb{R}$ to $\mathbb{C}$ with $L^1$-norm and convolution as algebra multiplication. Assume that we realized that all homomorphisms from $L$ to $\mathbb{C}$ are identical zero and Fourier transform values in specific real points: $f\rightarrow \int f(t)e^{it\alpha}dt$. We may add unity $e$ to $L$ artifically, just considering new Banach algebra $A:=L\oplus \mathbb{C}\cdot e$ with natural operations. Then the fact that any $L^1$-function with zero Fourier transform must be zero itself may be rephrased algebraically: algebra $A$ is semisimple (as maximal ideals of unital banach algebras correspond to homomorphisms to $\mathbb{C}$ by Gelfand-Mazur theorem).

My question is whether this may be proved a priori and independently (and maybe for some wide class of commutative unital Banach algebras).