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

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?

• I.G. Connell, On the group ring, Canad. J. Math. 15 (1963), 650–685 proved the group algebra is Artinian iff the group is finite. Zelmanov generalized this to semigroups. Semisimplicity is not needed. Jun 27 at 10:30

No: if the discrete group algebra $$\mathbb C G$$ is semisimple, then $$G$$ is finite.
Consider the trivial $$G$$-module $$\mathbb C$$ and the quotient map of $$G$$-modules $$\mathbb C G \to \mathbb C$$ which sends $$g \mapsto 1$$ for all $$g \in G$$. If $$\mathbb C G$$ is semisimple, then this map splits, giving a nonzero $$G$$-invariant element $$e = \sum_{g \in G} a_g g \in \mathbb C G$$ where $$a_g \in \mathbb C$$ are almost all zero and $$\sum_{g\in G} a_g = 1.$$ The element $$e$$ is invariant if and only if $$a_g = a_1$$ for all $$g \in G$$, hence such $$e$$ exists if and only if $$G$$ is finite.