An example of an infinite amenable group with the ISR property

Abstract

Let $G$ be $S_{\mathbb{N}}$, the finitary permutation (i.e. permutations with finite support) group on positive integers $\mathbb{N}$. We prove that $G$ has the invariant von Neumann subalgebras rigidity (ISR, for short) property as introduced in Amrutam-Jiang's work. More precisely, every $G$-invariant von Neumann subalgebra $P\subseteq L(G)$ is of the form $L(H)$ for some normal sugbroup $H\lhd G$ and in this case, $H=\{e\}, A_{\mathbb{N}}$ or $G$, where $A_{\mathbb{N}}$ denotes the finitary alternating group on $\mathbb{N}$, i.e. the subgroup of all even permutations in $S_{\mathbb{N}}$. This gives the first known example of an infinite amenable group with the ISR property.