Integrability of analytic almost complex structures on Banach manifolds

We prove that the classical integrability condition for almost complex structures on finite-dimensional smooth manifolds also works in infinite dimensions in the case of almost complex structures that are real analytic on real analytic Banach manifolds. With this result at hand, we extend some known results concerning existence of invariant complex structures on homogeneous spaces of Banach-Lie groups. By way of illustration, we construct the complex flag manifolds associated with unital C*-algebras.