Localization and projections on bi-parameter BMO

We prove that for any operator $T$ on bi–parameter BMO the identity factors through $T$ or $I-T$. Bourgain’s localization method provides the conceptual framework of our proof. It consists in replacing the factorization problem on the non–separable bi–parameter BMO by its localized, finite dimensional counterpart. We solve the resulting finite dimensional factorization problems by exploiting the geometry and combinatorics of colored dyadic rectangles.