I am confused with the underlined equation in the following picture.
[![enter image description here][1]][1]
I know that a *-isomorphism communicates with continuous functional calculus since every continuous functions on the compact subset can be  approximated uniformly by polynomials. 

To prove the underlined equation, we have to prove that $f\to \Phi(f(j_A(a)))$ is the Borel functional calculus on $\Phi(j_A(a))$.
Given a net $(f_\lambda)$ of continuous functions converging to $0$ respect to $\sigma(A^{**},A^*)$ topology , we have $f_\lambda(j_A(a))\xrightarrow{\sigma(A^{**},A^*)}0$, but how to obtain $\Phi(f_\lambda(j_A(a)))\xrightarrow{\sigma(B^{**},B^*)}0$?


  [1]: https://i.sstatic.net/5JGb9.png