# Natural map $C^*(G) \to M(A\rtimes G)$

Let $A$ be a C*-algebra, let $G$ be a locally compact group, and let $\alpha\colon G\to\mathrm{Aut}(A)$ be a (strongly) continuous action. It is well known that there is a natural map $\iota_G\colon C^*(G)\to M(A\rtimes_\alpha G)$.

Question: Is the natural map $\iota_G\colon C^*(G)\to M(A\rtimes_\alpha G)$ injective?

This is true for reduced group $C^*$-algebras and reduced crossed products, and I wonder whether it holds in the universal case too.

One would need to know something about what group representations $v\colon G\to \mathcal{U}(H_v)$ can be part of a covariant representation $(\pi,v)$ of the dynamical system $(G,A,\alpha)$ on $H_v$. My question can be restated as follows:

Reformulation: Let $w$ be the direct sum of all unitary representations $v\colon G\to \mathcal{U}(H_v)$ such that there exists a representation $\pi\colon A\to \mathcal{B}(H_v)$ so that $(\pi,v)$ is a covariant pair. Is $u$ the universal representation of $G$?

The answer is no. For a counter example, take an amenable action $\alpha$ of a non-amenable discrete group G on a unital C*-algebra $A$. Then $A\rtimes_\alpha G$ coincides with the reduced crossed product and hence the standard conditional expectation is faithful. If your map were injective, the standard conditional expectation on $C^*(G)$ would also be faithful, but it isn't.

• Perhaps I should add that, by a result of Ozawa (arXiv:math/0002185), every exact group, such as $F_2$, admits an amenable action on a compact space.
– Ruy
Sep 14 '16 at 21:16
• If I am not mistaken your argument shows that $\iota_G$ is never injective when $A$ is exact and $\alpha$ is an amenable action of a nonamenable group. It will also be interesting to know whether $\iota_G$ can be injective for nonamenable actions. Nov 23 '16 at 14:58
• @Adrian, if $A$ is unital, the argument works regardless of exactness. In the nonunital case I guess one would have to extend the conditional expectation to the multiplier algebra and, if this works, exactness of $A$ will not be needed. Can you explain your argument?
– Ruy
Nov 23 '16 at 18:58
• My apologies, @Ruy. I missread your argument. I just read the beginning and thought that you were using that if $A$ is exact and $\alpha$ amenable, $A \rtimes_\alpha G$ is exact and that $C^\ast G$ is not exact for nonamenable $G$. You cannot embbed a non exact algebra in an exact one. Your argument works fine since we are not considering all possible embeddings, just $\iota_G$. My bad. Nov 23 '16 at 20:50

No. Let $G$ be a locally compact group acting on a compact Hausdorff topological space $X$. Then the canonical $*$-homomorphism $C^*(G)\rightarrow C(X)\rtimes G$ is injective iff there is an invariant measure on $X$. The later can be equivalent to amenability of $G$ for some $X=\beta^{lu}(G)$. When $G$ is discrete, then $\beta^{lu}(G)$ is the Stone-Cech compactification of $G$.

• Isn't this just a special case of @Ruy's answer? Sep 22 '16 at 19:41
• @YemonChoi: I just want to indicate the converse implication. You can replace C(X) by a unital $C^*$-algebra $A$ and replace an invariant measure on $X$ by a $G$-invariant state on $A$ to obtain a generalization. Sep 24 '16 at 10:50