Symplectic leaves in real Banach Lie-Poisson spaces

We present several large classes of real Banach Lie-Poisson spaces whose characteristic distributions are integrable, the integral manifolds being symplectic leaves just as in finite dimensions. We also investigate when these leaves are embedded submanifolds or when they have K\"ahler structures. Our results apply to the real Banach Lie-Poisson spaces provided by the self-adjoint parts of preduals of arbitrary $W^*$-algebras, as well as of certain operator ideals.