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

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.

arXiv comment9 pages(v1), 7 pages(v2, v3); typos corrected, references added(v2); typos corrected, some expressions changed(v3)