Abstract
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.
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Acknowledgements
Y. J. is partially supported by National Natural Science Foundation of China (Grant No. 12001081) and the Fundamental Research Funds for the Central Universities (Grant No. DUT19RC(3)075). X. Z. is partially supported by National Natural Science Foundation of China (Grant No. 12001085) and the Scientific Research Fund of Liaoning Provincial Education Department (Grant No. LJKZ1047). We are grateful to Prof. Adam Skalski and Dr. Amrutam Tattwamasi for helpful comments. We also thank the anonymous referee for excellent comments which help improving the readability of the paper greatly.
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Jiang, Y., Zhou, X. An example of an infinite amenable group with the ISR property. Math. Z. 307, 23 (2024). https://doi.org/10.1007/s00209-024-03495-8
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DOI: https://doi.org/10.1007/s00209-024-03495-8