Let $\Gamma$ be a finitely generated group and let $A=\mathbb{C}[\Gamma]$ be the corresponding group algebra over $\mathbb{C}$. Let $X$ be the set of all maximal left ideals of $A$ and let $X_0=\{I \in X : \,\, A/I \textrm{ is finite dimensional } \mathbb{C} \textrm{-vector space}\}.$

Now consider $J= \bigcap_{I \in X} I$ and $J_0= \bigcap_{I \in X_0} I$. It is easy to see that $J$ and $J_0$ are two-sided ideals. The ideal $J$ is the Jacobson radical of $A$ and it has been proven that it is the zero ideal.

My question is the following: For which groups $\Gamma$ is $J_0$ also the zero ideal?

Is there some literature on this question? I am interested in necessary and sufficient criteria as well as in examples, but even more in non-examples.

What I know so far: $J_0$ is the zero ideal:

-if $\Gamma$ is finite (that's trivial),

-if $\Gamma$ is commutative,

-if $\Gamma$ is the free group in finitely many generators.