What does the representation theory of the reduced C*-algebra correspond to? - MathOverflow

What does the representation theory of the reduced C*-algebra correspond to?

2010-08-16T16:38:07Z

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.)

Answer by Owen Sizemore for What does the representation theory of the reduced C*-algebra correspond to?

2010-08-16T17:16:30Z

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.

Answer by Alain Valette for What does the representation theory of the reduced C*-algebra correspond to?

2011-05-06T20:22:00Z

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.