I am seeking for an Artin $k$-algebra (especially for group algebra) which is infinite-dimensional over some field $k$. It's known that any complex group algebra has trivial Jacobson radical. So I have the following question:


Is there a countable discrete infinite group $G$ over which the group algebra $\mathbb{C} G$ is semisimple?



Is there a countable discrete infinite group $G$ over which the group algebra $\mathbb{C} G$ is semisimple?
The answer is negative as any semisimple Hopf algebra is finite-dimensional. More generally, the same conclusion is true for all Artinian Hopf algebras. See e.g. [Liu and Zhang - Artinian Hopf algebras are finite dimensional](https://www.ams.org/journals/proc/2007-135-06/S0002-9939-07-08711-4/home.html).


