  We consider a Strang-type second order operator-splitting discretization for
the Cahn-Hilliard equation. We introduce a new theoretical framework and prove
uniform energy stability of the numerical solution and persistence of all
higher Sobolev norms. This is the first strong stability result for second
order operator-splitting methods for the Cahn-Hilliard equation. In particular
we settle several long-standing open issues in the work of Cheng, Kurganov, Qu
and Tang \cite{Tang15}.
