# clarification on the definition of a group C*-algebra

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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First of all I'd restrict to discrete groups first for making things a bit easier (I guess you don't want a general topological group but rather a locally compact one - to guarantee that $\|a\|$ is actually a norm). Then note that $\|\pi_{u}(a)\|$ defins a semi-norm on $\mathbb{C}G$ for any $\pi$. Now it makes sense to speak of all semi-norms that arise in this way. If there are sufficiently many representations (that's why you need some sort of restriction on $G$) to send $a$ to some nonzero element in $\operatorname{End}(A)$ then $\|a\|$ is non-zero. This defines a norm on $\mathbb{C}G$ – t.b. May 20 '11 at 14:27
and you complete $\mathbb{C}G$ with respect to that norm, which is a straightforward process. In the locally compact case you have to work quite a bit in order to see that there are sufficiently many representations, the compact case is a bit easier because of Peter-Weyl. I think all this is quite well explained in Bekka-de la Harpe-Valette's book on property (T) in an appendix (at least for second countable groups). Otherwise the only really detailed reference I know is Dixmier's book on $C^{\ast}$-algebras. As we have some operator-theorists here, I let them jump in for more details. – t.b. May 20 '11 at 14:31
Two other references: Pedersen, $C^{\ast}$-algebras and their automorphism groups and Takesaki's Theory of operator algebras but I don't know which volume, off-hand (I guess most of this is in the exercises in volume one but volume two might have more information) – t.b. May 20 '11 at 14:35
Finally, as any $C^\ast$-algebra has a faithful (hence isometric) representation on some (possibly humungous) Hilbert space by the GNS construction, it's actually enough to look at unitary representations on Hilbert spaces in the first place. The size of $G$ cuts down the possible size of that Hilbert space on which G acts unitarily and non-trivially, and thus the supremum in the definition isn't really problematic. – t.b. May 20 '11 at 14:57
Thanks so much for the speedy response! I'll look up some of those references and get back here if I'm still confused. – Jon Paprocki May 20 '11 at 15:08