Vol 11, No 1 (2019)

The present paper is a brief survey of properties of finite ultrametric spaces X and corresponding properties of the representing trees T_X obtained by authors over the last six years. Some new results are also presented. In particular, a structural characteristic of the representing trees T_X is found for the finite ultrametric spacesX which admit a ball-preserving mapping f: Y → Z for all nonempty Y ⊆ X and Z ⊆ Y.

This work will be centered in commutative Banach subalgebras of the algebra of bounded linear operators defined on free Banach spaces of countable type. The main goal of this work will be to formulate a representation theorem for these operators through integrals defined by spectral measures type. In order to get this objective, we will show that, under special conditions, each one of these algebras is isometrically isomorphic to some space of continuous functions defined over a compact set. Then, we will identify such compact sets developing the Gelfand space theory in the non-Archimedean setting. This fact will allow us to define a measure which is known as spectral measure. As a second goal, we will formulate a matrix representation theorem for this class of operators in which the entries of the matrices will be integrals coming from scalar measures.

We show that the Connes-Marcolli GL₂-system can be represented on the Big Picture, a combinatorial gadget introduced by Conway in order to understand various results about congruence subgroups pictorially. In this representation the time evolution of the GL₂-system is implemented by Conway’s distance between projective classes of commensurable lattices. We exploit these results in order to associate quantum statistical mechanical systems to congruence subgroups. This work is motivated by the study of congruence subgroups and their principal moduli in connection with monstrous moonshine.