states in the cross product $C^*$-algebras

Question

Let $\tau$ be a state of cross product $C^*$-algebra $A\rtimes G$, where $A$ is a $C^*$-algebra and $G$ is a discrete group. Is $\tau $ also a state of $A$?

Answer 1

Yes. Let $\phi:A\hookrightarrow A\rtimes G$ denote the canonical embedding, and let $\tau_A=\tau\circ\phi$, so that $\tau_A$ is the restriction in question. As a composition of two positive linear maps, we have that $\tau_A$ is positive and linear. To show that $\tau_A$ is a state, let $(u_\lambda)$ be an approximate unit for $A$. Then $(\phi(u_\lambda))$ is an approximate unit for $A\rtimes G$, so we have $$\|\tau_A\|=\lim_\lambda\tau_A(u_\lambda)=\lim_\lambda\tau(\phi(u_\lambda))=\|\tau\|=1$$ and therefore $\tau_A$ is a state.