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](https://mathoverflow.net/questions/23408/reference-for-the-gelfand-neumark-theorem-for-commutative-von-neumann-algebras)
2) [What kind of completion is this?](https://mathoverflow.net/questions/60328/what-kind-of-completion-is-this)



Ultraweak topology in abelian von Neumann algebras
მამუკა ჯიბლაძე hints that "hyperstonian" is an important definition here. I have struggled to find a good internet reference, and so am following Section 1 of Chapter III of Takesaki's book.


A *Stonian* space is a compact Hausdorff [Extremally disconnected space](https://en.wikipedia.org/wiki/Extremally_disconnected_space). A *rare* or [nowhere dense](https://en.wikipedia.org/wiki/Nowhere_dense_set) set $M$ is such that the closure of $M$ has empty interior.


Let $X$ be a Stonian space and let $C\_{\mathbb R}(X)$ be the space of real-valued continuous functions on $X$. A (positive, Radon) measure $\mu$ on $X$ is *normal* if whenever $(f\_i)$ is an increasing bounded net in $C\_{\mathbb R}(X)$ and $f$ is the least upper bound of $\{f\_i\}$ in $C\_{\mathbb R}(X)$ (which exists as $X$ is Stonian), then
$$ \int\_X f \ d\mu = \sup\_i \int\_X f\_i \ d\mu. $$
Alternatively, $\mu$ is normal if $\mu(M)=0$ for all (closed) rare sets $M$.


Finally, $X$ is *hyperstonian* if it admits sufficiently many normal measures: for any non-zero $f\in C\_{\mathbb R}(X)$ there is a normal $\mu$ with $\int\_X f\ d\mu\not=0$.


For a compact Hausdorff $X$, we have that $C(X)$ is a von Neumann algebra exactly when $X$ is hyperstonian.


Define a real-valued Radon measure to be normal if its positive and negative parts are normal; analogously for complex Radon measures. Following the proofs through, it follows that the predual of $C(X)$ is exactly the collection of complex normal measures on $X$. 


But I wonder if this answers the original question: is this in terms of "$X$ and its topology"? However, notice that the difference between Stonian and Hyperstonian is very measure theoretic in nature.


