Celebratio Mathematica — Jones

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Works connected to Masamichi Takesaki

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V. F. R. Jones and M. Takesaki: “Actions of compact abelian groups on semifinite injective factors,” Acta Math. 153 (1984), pp. 213–258. MR 766264 Zbl 0588.46042 article

The Clifford algebra construction allows one to associate with any unitary representation of a group an action of the group on an algebra in a functorial way. If the representation is infinite dimensional one must allow infinite dimensional algebras and one is led immediately to consider actions of groups on factors not of type \( \mathrm{I} \). Using this approach, Blattner showed in [1958] that any separable locally compact group has a faithful action, by outer automorphisms and the identity, on the hyperfinite type \( \mathrm{II}_1 \) factor \( \mathscr{R} \). It was certainly not obvious at the time that one might hope to say much more about the actions even of finite cyclic groups on \( \mathscr{R} \), but largely thanks to work of Connes, much progress has been made. This paper adds another step in a continuing project by giving a detailed description of all actions of a compact abelian group on semifinite injective factors.