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 $\iota\_G\colon C^\*(G)\to M(A\rtimes\_\alpha G)$. 




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**Question:** Is the natural map $\iota\_G\colon C^\*(G)\to M(A\rtimes\_\alpha G)$ injective?




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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: 




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**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 $G$? 




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


