Homotopy decompositions of stable gauge groups

There is widespread interest in the gauge groups of principal SU(n)-bundles over simply-connected four-manifolds, and the gauge groups of principal U(n)-bundles over compact, orientable surfaces. We show that in the stable case of SU(∞) (resp. U(∞)), the gauge groups decompose, up to a multiplicative homotopy which holds integrally, as an explicit product where each factor is an iterated loop space of SU(∞) (resp. U(∞)).