Question

Is there an example of an uncountable Boolean algebra $B$ in which every chain is countable and such that $\ell_\infty$ embeds into the Banach space $C(\mbox{Stone }B)$? The latter requirement is not very important, I just want to exclude some trivial cases, like the algebra of finite/cofinite subsets on some uncountable set.

Answer 1

The second requirement is too strict: it makes the first one impossible.

The commutative von Neumann algebra $\ell^\infty(\mathbb{N})$ has as Gelfand spectrum the Stone-Cech compactification of $\mathbb{N}$ (with the discrete topology). This, in turn, is the Stone space of the Boolean algebra $\mathcal{P}(\mathbb{N})$, the powerset of the natural numbers. So $\ell^\infty(\mathbb{N}) \cong C(\mathop{Stone}(\mathcal{P}(\mathbb{N})))$ embeds in $C(\mathop{Stone}(B))$ if and only if $\mathcal{P}(\mathbb{N})$ embeds in $B$. But the former has uncountable chains. So if $B$ satisfies the second requirement, it has uncountable chains, and cannot satisfy the first requirement.