  To 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.
