Group C* algebra realized as crossed product

Question

Can group $C^{\ast}$ algebras be realized as crossed product? For instance, do we need extra property on group?

Answer 1

Sure. Let $\Gamma$ be a discrete group. Then $$C_u^*(\Gamma)\cong \mathbb{C}\rtimes \Gamma, \quad C_r^*(\Gamma) \cong \mathbb{C}\rtimes_r \Gamma$$ where $\Gamma$ acts on $\mathbb{C}$ trivially (in fact, there is only one action of $\Gamma$ on $\mathbb{C}$).