Monod, Nicolas2020-08-152020-08-152020-08-152020-08-0310.1007/s00208-020-02034-0https://infoscience.epfl.ch/handle/20.500.14299/170860WOS:000554840400001Every Gelfand pair (G, K) admits a decomposition G = K P, where P < G is an amenable subgroup. In particular, the Furstenberg boundary of G is homogeneous. Applications include the complete classification of non-positively curved 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.Mathematicstext::journal::journal article::research article