2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/111434To each regular affine building there is naturally associated a commutative algebra A spanned by vertex set averaging operators. In this paper we study the algebra homomorphisms from A into the complex numbers. In particular, we provide two explicit formulae for these homomorphisms; one in terms of the Macdonald spherical functions, and another as an integral over the boundary of the building. We also determine the associated Plancherel measure, and we compute the l^2-operator norms of each basis element of A.To appear in Mathematische ZeitschriftFunctional Analysis20E42; 43A90text