An example of an infinite amenable group with the ISR property

Let G be \(S_{\mathbb {N}}\), the finitary permutation (i.e., permutations with finite support) group on the set of 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 subgroup \(H\lhd G\) 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.