Let $\Gamma$ be a non-amenable discrete group and consider its Furstenberg boundary $\partial_F \Gamma$. It is known that this is a compact topological space which is stonean (equivalently: extremely disconnected). The function algebra $C(\partial_F \Gamma)$ is injective as a $C^*$-algebra, and thus an $A$-$W^*$-algebra. In particular, $C(\partial_F \Gamma)$ is in some sense 'close' to being a $W^*$-algebra.

Questions:

(1) Is $C(\partial_F \Gamma)$ ever a $W^*$-algebra?

(2) Assume $\Gamma$ is a discrete group so that $(1)$ has a positive answer: we can consider $C(\partial_F \Gamma)\subseteq \ell^\infty(\Gamma)$. When is it true that this is an inclusion of von Neumann algebras?

Let $\Gamma$ be a non-amenable discrete group and consider its Furstenberg boundary $\partial_F \Gamma$. It is known that this is a compact topological space which is stonean (equivalently: extremely disconnected). The function algebra $C(\partial_F \Gamma)$ is injective as a $C^*$-algebra, and thus an $A$-$W^*$-algebra. In particular, $C(\partial_F \Gamma)$ is in some sense 'close' to being a $W^*$-algebra.

Questions:

(1) Is $C(\partial_F \Gamma)$ ever a $W^*$-algebra? Equivalently, is $\partial_F\Gamma$ ever a hyperstonean space?

(2) Assume $\Gamma$ is a discrete group so that $(1)$ has a positive answer: we can consider $C(\partial_F \Gamma)\subseteq \ell^\infty(\Gamma)$. When is it true that this is an inclusion of von Neumann algebras?