C*-algebras of discrete groups

We first recall some classical notation from noncommutative Abstract Harmonic Analysis on an arbitrary discrete group G. We denote by e (and sometimes by eG) the unit element. Let π :G → B(H) be a unitary representation of G. We denote by C∗ π (G) the C∗-algebra generated by the range of π . Equivalently, C∗ π (G) is the closed linear span of π(G). In particular, this applies to the so-called universal representation of G, a notion that we now recall. Let (πj )j∈I be a family of unitary representations of G, say πj :G→ B(Hj ) in which every equivalence class of a cyclic unitary representation ofG has an equivalent copy. Now one can define the “universal” representation UG :G→ B(H) of G by setting UG = ⊕j∈I πj on H = ⊕j∈IHj .