Let $G$ be locally compact group. Define group algebra as $$L^1(G)=\{f\colon G\to\Bbb{C}\mid\int\lvert f(x)\rvert\, dx<\infty\}$$ with convolution product. When is the group algebra $L^1(G)$ semisimple?
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Always, according to Naĭmark, Normed Algebras, VII p. 380. 


I found my answer also in corollary 4.34 page 103 of a book which was named "A course in abstract harmonic analysis" By Folland. In there it was proven that for any locally compact commutative group $G$ the group algebra $L^1(G)$ and measure algebra $M(G)$ are semisimple. 

