The Mazur-Ulam property for commutative von Neumann algebras

Let $(\Omega,\mu)$ be a $\sigma$-finite measure space. Given a Banach space $X$, let the symbol $S(X)$ stand for the unit sphere of $X$. We prove that the space $L^{\infty} (\Omega,\mu)$ of all complex-valued measurable essentially bounded functions equipped with the essential supremum norm, satisfies the Mazur-Ulam property, that is, if $X$ is any complex Banach space, every surjective isometry $\Delta: S(L^{\infty} (\Omega,\mu))\to S(X)$ admits an extension to a surjective real linear isometry $T: L^{\infty} (\Omega,\mu)\to X$. This conclusion is derived from a more general statement which assures that every surjective isometry $\Delta : S(C(K))\to S(X),$ where $K$ is a Stonean space, admits an extension to a surjective real linear isometry from $C(K)$ onto $X$.