Let $S$ be a set, $\Sigma$ an algebra on $S$ and consider the Banach space $B(S, \Sigma)$ of bounded $\Sigma$-measurable functions $f : S \to \mathbb{R}$. Denote by $K_\Sigma$ the Gelfand spectrum of $B(S, \Sigma)$, so that $B(S, \Sigma)$ can be identified with $C(K_\Sigma)$. Under what conditions on $\Sigma$ is $K_\Sigma$ Stonean, i.e. an extremally disconnected compact Hausdorff space? In particular, I am interested in the following case: Is $K_\Sigma$ Stonean if $S$ is compact Hausdorff and $\Sigma$ the Borel or Baire $\sigma$-algebra?
Motivation: If $B(S, \Sigma)$ is Stonean, then $C(K_\Sigma)$ is a Grothendieck space, meaning that weak$^*$ and weak convergence for sequences in $B(S, \Sigma)' = ba(\Sigma)$ coincide.