I'm learning about representations of groups and their group C-algebras and I am trying to understand the relationship between bounded representations of a group and its group C-algebra.
Let $G$ be a discrete group. A bounded representation of $G$ is a homomorphism $\pi: G \to B(H)$ where $H$ is a Hilbert space, and such that $|\pi| = \sup_{g \in G} \|\pi(g)\| < \infty$. We call $\pi$ unitary if $\pi(g)$ is unitary for all $g \in G$. We define the group C-algebra of $G$ as follows. The group algebra of $G$ is $\mathbb C [G] = \bigoplus_{g \in G} \mathbb C$ with convolution for multiplication, and involution given by $(\sum c_g g)^\ast = \sum \overline{c_g} g^{-1}$. We define a norm on $\mathbb C[G]$ by $$\|\sum c_g g\|_\ast = \sup\{\|\sum c_g \pi(g)\|: \pi\text{ is a unitary representation of $G$}\}$$ The completion of $\mathbb{C}[G]$ with respect to this norm is a C-algebra called the group C-algebra of $G$ and is denoted $C^\ast(G)$. If $A$ is a C-algebra then a bounded representation of $A$ is a bounded algebra homomorphism $\Gamma: A \to B(H)$. We say $\Gamma$ is a unitary representation if $\Gamma(a)$ is unitary for all $a \in A$.
If $\pi:G \to B(H)$ is a unitary representation, then we can define a map $\tilde \pi: \mathbb{C}[G] \to B(H)$ by $$\tilde \pi(\sum c_g g) \mapsto \sum c_g \pi(g)$$ Then from the definition of $\| \cdot \|_\ast$ we see that $\tilde \pi$ is bounded and hence extends to a bounded map on $C^\ast(G)$. Moreover unitarity also shows that $\pi$ is a $\ast$-homomorphism. Conversely if $\Gamma: C^\ast(G) \to B(H)$ is a unitary representation then $\Gamma \vert _G$ is a unitary representation of $G$. Thus there is a 1-1 correspondence between unitary representations of $G$ and $\ast$-representations of $C^\ast(G)$.
Now with this in mind, I am trying to figure out if there is any relationship between bounded representations of $G$ and bounded representations of $C^\ast(G)$. Suppose $\pi:G \to B(H)$ is a bounded representation. We can still define $\tilde \pi: \mathbb{C}[G] \to B(H)$ as above, but I have no clue whether or not $\tilde \pi$ will be bounded. Similarly, if $\Gamma: G \to B(H)$ is a bounded representation then $\Gamma \vert G$ is a homomorphism of $G$ onto $B(H)$, but I again have no idea on how to tell if it is bounded or not.
Any help is appreciated!