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?



When is the group algebra $L^1(G)$ semisimple?
Always, according to Naĭmark, *Normed Algebras*, VII p. 380.


