Actions of finite groups on finite von Neumann algebras and the relative entropy (Q1098387)

M being a finite Von Neumann algebra and M G its fixed point subalgebra under the action \(\alpha\) of a finite group G, let H(M\(| M\) G) be the corresponding relative entropy in the sense of \textit{A. Connes}-\textit{E. Störmer} [Acta Math., 134, 289-306 (1975; Zbl 0326.46032)] and \textit{Pimsner}-\textit{Popa} [Ann. Sci. Ecole Norm. Sup. 19, 57-106 (1986)]. The authors, by reducing to the case where \(\alpha\) is centrally ergodic and M is a factor, give a formula for the computation of H(M\(| M\) G). In particular they prove that H(M\(| M\) G)\(\leq \log | G|\). Their method relies on Jones' conjucagy invariants of the action \(\alpha\) [\textit{V. F. R. Jones}, Mem. Am. Math. Soc. 237, 70 p. (1980; Zbl 0454.46045)].