Some Properties of C*-Algebras Associated to Discrete Linear Groups

Let Γ be a discrete (countable) group. There are two distinguished C*-algebras one may associate to Γ : the reduced C*-algebra C* _r (Γ) which is the norm closure of the linear span of {(λ (γ) : γ ∈ Γ}, where λ is the left regular representation of Γ on ℓ2 (Γ), defined by $$lambda (\gamma)f(x) = f({\gamma ^{ - 1}}x)\forall \gamma ,{\text{x}} \in \Gamma ,f \in {\ell ^2}(\Gamma);$$the maximal (or full) C* -algebra C*(Γ) which is the completion of the group algebra ℂΓ for the C*-norm $$parallel \sum {{c_\gamma }\gamma {\parallel _{{\text{max}}}} = {\text{sup}}\left\{ \parallel \right.} \sum {{c_\gamma }\pi (\gamma )\parallel :\pi } {\text{ is a}}*{\text{-representation of }}\mathbb{C}\left. \Gamma \right\}$$.