Even Automorphisms of Trees and Inductive Algebras

Let (pi; H) be a unitary representation of a locally compact group G. A com-
mutative subalgebra A of B(H) is called inductive when pi(g)A pi(g^(-1)) = A
for all g. The classification of maximal inductive algebras sheds light on the possible realizations of on function spaces. In this paper we deal with the automorphism group of a locally finite homogeneous tree and its principal series spherical
representations. We show that for some exceptional representations there exists
just one inductive algebra besides those known. Finally, we generalize the main
results to the subgroup of even automorphisms of the tree.