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Let $G$ be a locally compact group. The group C*-algebra $C^* (G)$ is designed to come with a natural bijection between its (nondegenerate) representations and the (strongly continuous, unitary) representations of $G$.

Question: Is there a similar statement for the reduced group C*-algebra $C^*_r (G)$?

If the answer is no, I'll probably end up asking for the actual purpose of defining $C^*_r (G)$. So far, I know that its isomorphic to $C^* (G)$ in important cases, and that its construction is in some sense simpler than the one of $C^* (G)$.

(The definitions and the claims used above can be found in Blackadar's Operator Algebras.)

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One glib answer is that the left regular representation $\lambda$ of a locally compact group has something of a privileged position among all unitary representations: it is arguably one of the most natural ones to consider; and tensoring it with any other (cts) unitary representation will give a representation equivalent to an amplification of $\lambda$ (this is Fell's absorption principle). So the study of the C*-algebra generated by $\lambda$, namely $C^*_r(G)$, is a reasonable enterprise. –  Yemon Choi Aug 16 '10 at 19:33
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Another glib point: when $G$ is nonamenable the full group C*-algebra is often thought of as being "very big", in the sense that it is not clear how to get at the elements arising from completion of $L^1(G)$. This befits its status as a universal object. –  Yemon Choi Aug 16 '10 at 19:40
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2 Answers

up vote 5 down vote accepted

So there is a similar property.

Now $C^*_r(G)$ is the $C^\star$-algebra generated by the left-regular rep. It a general theorem that if you have a unitary rep $\pi:G\rightarrow \mathcal{U} (H)$, and if $\rho: G\rightarrow \mathcal{U}(K)$ is another unitary rep that is weakly contained ($\rho\prec\pi$) in $\pi$, then there is a surjective map from the reduced $C^\star$-algebra to the algebra generated by $\rho(G)\subset B(K)$

So $C^\star_r(G)$ surjects onto all reps that weakly contain the left-regular.

Note: $C^\star_r(G)\simeq C^\star(G)$ iff G is amenable.

A good source for most of this

http://perso.univ-rennes1.fr/bachir.bekka/KazhdanTotal.pdf

This is the pdf of a book about Property (T). Appendix F.4 is about the above questions but the whole book is of interest for people in operator algebras, representation theory, geometric group theory, and many other fields.

EDIT: Another good source, which is directed to Yemon's comment is http://arxiv.org/PS_cache/math/pdf/0509/0509450v1.pdf

This is a survey, by Pierre de la Harpe, of groups whose reduced $C^\star$-algebra is simple.

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As an aside, which you didn't mention but might be of interest to the original questioner: it is perfectly possible for the reduced group C*_algebra to be simple, i.e. have no non-trivial closed ideals. This holds for the free group on $n$ generators, for instance; and so the "abelianization representation" of the free group does not come from a representation of the reduced group C*-algebra. –  Yemon Choi Aug 16 '10 at 19:36
    
@Yemon Choi: Thank you for your interesting comments. Could you please explain what you mean with "abelianization representation"? Is there some obvious representation of $\mathbb Z^n$ that you want to compose with the "abelianization homomorphism" $F_n\to\mathbb Z^n$? –  Rasmus Bentmann Aug 17 '10 at 10:03
    
@Rasmus: my apologies for being unclear in my haste. I really meant "any representation of $F_n$ which factors through the abelianization homomorphism $F_n\to ${\mathbb Z}^n$" -- which is not what I originally wrote! –  Yemon Choi Aug 18 '10 at 5:33
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For suitable locally compact groups $G$ (separable, unimodular, type I), there is a measure $\mu$ on the dual $\hat{G}$ such that, for every function $f\in L^1(G)\cap L^2(G)$: $$\int_G |f(g)|^2dg=\int_{\hat{G}}\|\pi(f)\|^2_{HS}d\mu(\pi)$$ where $\|.\|_{HS}$ denotes the Hilbert-Schmidt norm. The measure $\mu$ is the Plancherel measure of $G$ and its support is exactly the reduced dual, i.e. the dual of $C^*_r(G)$. For all this, see section 18.8 in J. Dixmier, $C^*$-algebras, North Holland, 1977.

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