Every Gelfand pair (G,K) admits a decomposition G=KP, where P<G is an amenable subgroup. In particular, the Furstenberg boundary of G is homogeneous.
Applications include the classification of non-positively curved spaces admitting Gelfand pairs, relying on earlier joint work with Caprace, as well as a canonical family of pure spherical functions in the sense of Gelfand–Godement for general Gelfand pairs.