Let $A$ be a C*-algebra, let $G$ be a 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 $G$?

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 $G$?