Jones' conjecture on outer automorphisms of group von Neumann algebras

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Let GG be an ICC group with Kazhdan's property (T). Write Char⁡(G)\operatorname{Char}(G) for the group of characters of GG, Out⁡(G)\operatorname{Out}(G) for its outer automorphism group, and Out⁡(L⁡(G))\operatorname{Out}(\operatorname{L}(G)) for the outer automorphism group of its group von Neumann algebra. Jones' conjecture. One has

Out⁡(L⁡(G))≅Char⁡(G)⋊Out⁡(G).\operatorname{Out}(\operatorname{L}(G))\cong \operatorname{Char}(G)\rtimes \operatorname{Out}(G).

This asserts that property (T) makes the natural homomorphism from Char⁡(G)⋊Out⁡(G)\operatorname{Char}(G)\rtimes\operatorname{Out}(G) onto Out⁡(L⁡(G))\operatorname{Out}(\operatorname{L}(G)) surjective. The paper confirms the conjecture for a wide class of ICC groups with property (T), while the general statement remains open.

References

Primary source

I. Chifan, A. Ioana, D. Osin and B. Sun, “Wreath-like products of groups and their von Neumann algebras II: Outer automorphisms”, arXiv:2304.07457 (2025).

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