Function algebra of Furstenberg boundary $\partial_F \Gamma$: when is it a $W^*$-algebra?

Question

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?

Answer 1

The boundary $\partial_F\Gamma$ is never hyper-stonean for a (countable) non-amenable $\Gamma$. In fact, any action of any countable group $\Gamma$ on any infinite-dimensional (resp. non-atomic) abelian von Neumann algebra $A$ is not minimal (resp. topologically transitive) on the Gelfand spectrum $\hat{A}$ of $A$. (BTW, this answers Question 2.12 in M. Babillot, An introduction to Poisson boundaries of Lie groups.)

Proof. Take a singular state $\phi$ on $A$ (resp. the point evaluation at a point in $\hat{A}$) and consider the corresponding probability measure $\mu_\phi$ on $\hat{A}$. Enumerate $\Gamma=\{ g_n \}_{n=1}^\infty$ and consider the state $\psi := \sum_n 2^{-n} g_n\phi$. Since $\psi$ is still singular, there is a non-zero projection $p$ in $A$ such that $\psi(p) = 0$. The non-empty clopen subset $U \subset \hat{A}$ corresponding to $p$ satisfies $\mu_\phi( gU ) = 0$ for all $g$. Hence, $\bigcup_g gU$ is a non-trivial open $\Gamma$-invariant subset of $\hat{A}$.