Let $A$ be a $\mathrm C ^\ast$-algebra. It is a well known result that $A^{\ast \ast}$ is isometrically isomorphic as Banach space to $A^{''} = \pi_U(A)^{''}$, where $\pi_U$ is the universal representation. Furthermore, the isomorphism is the identity on $A$ (after viewing $A$ inside $\pi_U(A)$ resp. $A^{\ast \ast}$).
If we have an increasing bounded net in $A$, say $a_\lambda$, then the $a_\lambda$ converge strongly to some $g \in A^{''}$.
My question is the following: How does the strong convergence translate to convergence of the $a_\lambda$ to $g \in A^{\ast \ast}$ ? I.e. in what sense converges $a_\lambda$ to $g$ in $A^{\ast \ast}$ ?
The topologies that have a natural correspondence are the $\sigma$-weak on $A''$ and the weak$^*$ on $A^{**}$. If $a_\lambda\to g$ strongly in $A''$, it will also converge $\sigma$-weakly (convergence given by the normal states), and so it will converge in the weak$^*$-topology in $A^{**}$.