The Algebras Generated by the Laplace Operators in a Semi-homogeneous Tree

Abstract

In a semi-homogeneous tree, the set of edges is a transitive homogeneous space of the group of automorphisms, but the set of vertices is not (unless the tree is homogeneous): in fact, the latter splits into two disjoint homogeneous spaces V +, V − according to the homogeneity degree. With the goal of constructing maximal abelian convolution algebras, we consider two different algebras of radial functions on semi-homogeneous trees. The first consists of functions on the vertices of the tree: in this case the group of automorphisms gives rise to a convolution product only on V + and V − separately, and we show that the functions on V +, V − that are radial with respect to the natural distance form maximal abelian algebras, generated by the respective Laplace operators. The second algebra consists of functions on the edges of the tree: in this case, by choosing a reference edge, we show that no algebra that contains an element supported on the disc of radius one is radial, not even in a generalized sense that takes orientation into account. In particular, the two Laplace operators on the edges of a semi-homogeneous (non-homogeneous) tree do not generate a radial algebra, and neither does any weighted combination of them. It is also worth observing that the convolution for functions on edges has some unexpected properties: for instance, it does not preserve the parity of the distance, and the two Laplace operators never commute, not even on homogeneous trees.