Groups of isometries of ultrametric Urysohn spaces and their unitary representations

Abstract

We consider groups $\mathbb{I}$ of isometries of ultrametric Urysohn spaces $\mathbb{U}$. Such spaces $\mathbb{U}$ admit transparent realizations as boundaries of certain $R$-trees and the groups $\mathbb{I}$ are groups of automorphisms of these $R$-trees. Denote by $\mathbb{I}[X]\subset \mathbb{I}$ stabilizers of finite subspaces $X\subset \mathbb{U}$. Double cosets $\mathbb{I}[X]\cdot g\cdot \mathbb{I}[Y]$, where $g\in \mathbb{I}$, are enumerated by ultrametrics on union of spaces $X\cup Y$. We construct natural associative multiplications on double coset spaces $\mathbb{I}[X]\backslash \mathbb{I}/\mathbb{I}[X]$ and, more generally, multiplications $\mathbb{I}[X]\backslash \mathbb{I}/\mathbb{I}[Y]\,\times\, \mathbb{I}[Y]\backslash \mathbb{I}/\mathbb{I}[Z]\to \mathbb{I}[X]\backslash \mathbb{I}/\mathbb{I}[Z]$. These operations are a kind of canonical amalgamations of ultrametric spaces. On the other hand, this product can be interpreted in terms of partial isomorphisms of certain $R$-trees (in particular, we come to an inverse category). This allows us to classify all unitary representations of the groups $\mathbb{I}$ and to prove that groups $\mathbb{I}$ have type $I$. We also describe a universal semigroup compactification of $\mathbb{I}$ whose image in any unitary representation of $\mathbb{I}$ is compact.