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I've been trying to understand the definition of a group C*-algebra. Given a topological group $G$ and a C*-algebra $A$, let $u: G \to A$ define a unitary representation $U(G)$ of $G$ on $U(A)$, the unitary group of $A$. Let $\mathbb{C}G$ denote the group algebra of $G$. Then $u$ induces a homomorphism $\pi_u: \mathbb{C}G \to A$. We define the group C*-algebra of $G$ to be the completion of $\mathbb{C}G$ with respect to the norm

$\| a \| := \sup \{ \| \pi_u(a) \|: u: G \to U(A) \text{ is a homomorphism} \}$

I have a difficult time understanding what exactly completion with respect to this norm means. After thinking about it for a day or so, the only answer I've come up with is that given any Cauchy sequence $\{ x_n \}$ in $U(A)$ (where $A$ can be any C*-algebra and $U(A)$ can be any unitary representation), there is a corresponding Cauchy sequence $\{ y_n \}$ in $\mathbb{C}G$ and the group C*-algebra $C^*(G)$. In $C^*(G)$, we are guaranteed that the limit of this Cauchy sequence exists. Is this correct?

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    Thus, the group C*-algebra of say, $\mathbb{Z}$, is precisely the continuous functions on the unit circle, since unitary representations of $\mathbb{Z}$ are generated by$a$single unitary. I am still a bit foggy on the connection to the norm, but I think I'm satisfied for the time being.2011-05-26

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