On the type of the von Neumann algebra of an open subgroup of the Neretin group

TL;DR

This paper proves that a specific open subgroup of the Neretin group, which acts on a tree, is not of type I, providing an alternative proof to a recent result about the entire Neretin group.

Contribution

It demonstrates that the open subgroup of the Neretin group is not of type I, offering a new proof of the non-type I property of the Neretin group.

Findings

The open subgroup $ ext{O}_{d,k}$ is not of type I.

The Neretin group itself is not of type I.

Provides an alternative proof to recent results.

Abstract

The Neretin group $\mathcal{N}_{d,k}$ is the totally disconnected locally compact group consisting of almost automorphisms of the tree $\mathcal{T}_{d,k}$. This group has a distinguished open subgroup $\mathcal{O}_{d,k}$. We prove that this open subgroup is not of type I. This gives an alternative proof of the recent result of P.-E. Caprace, A. Le Boudec and N. Matte Bon which states that the Neretin group is not of type I, and answers their question whether $\mathcal{O}_{d,k}$ is of type I or not.