Let $A$ be an abelian von Neumann algebra acting on the (not necessarily separable) Hilbert space $\mathcal{H}$ (with identity $I$). From the Gelfand-Neumark theorem, there is a compact Hausdorff space $X$ such that $A \cong C(X)$, the $*$-algebra of complex-valued continuous functions on $X$. The space $X$ is in fact extremally disconnected.

Here goes my question(s). Just as the norm topology of $A$ is captured by the $\sup$ norm of the function algebra over $X$, what is an analogous description of the ultraweak topology on $A$? For instance, if we want to say $f_{\alpha} \rightarrow f$ in the ultraweak topology, how do we phrase such a statement in purely topological terms (in terms of $X$ and its topology)?

Thank you.

Edit: Clarified some aspects to make the question more specific. Looking around mathoverflow brought me to the following two discussions which captures the spirit of my question: 1) Reference for the Gelfand-Neumark theorem for commutative von Neumann algebras 2) What kind of completion is this?