If $\mathcal{A}$ is a $C^*$-Algebra, $\hat{\mathcal{A}}=spec(\mathcal{A})$, $(H_b,\pi_b)$ is any irreducible representation in equivalence class of $b$. I want to know where can i find proof of these theorems.
- $ \zeta : \mathcal{A} \rightarrow z \mathcal{A}^{**},~~ a\rightarrow z\bar{a}$
where $\bar{a}$ is an image of $a \in \mathcal{A}$ by natural embedding of $\mathcal{A}$ in it's second dual and $z$ is central projection in $\mathcal{A}^{**}$. Prove that $ \zeta$ is $C^*$-injective
$\mathsf{Z} :z \mathcal{A}^{**} \rightarrow \prod_{b~ \in \mathcal{\hat{A}}} B(H_b),~~~ \mathsf{Z}(a)= \prod _{b~ \in \mathcal{\hat{A}}} (\pi^{**} _b(a))$.
Where $\pi^{**} _b$ is $\sigma$-weak extension of $\pi_b$ to $\mathcal{A}^{**}$. Prove that $\mathsf{Z}$ is an isomorphism.