Mappings of Operator Algebras pp 233-238 | Cite as

# Relative Entropy of a Fixed Point Algebra

Chapter

## Abstract

The relative entropy *H*(*M*∣*N*) for a pair *N* ⊂*M* of finite von Neumann algebras was introduced and studied by M. Pimsner and S.Popa in [7]. One of their important results was to clarify the relationship between *H*(*M*∣*N*) and the Jones index [*M* : *N*] for a pair of finite factors ([2]). On the other hand, in [1], V. Jones succeeded in classifying actions of a finite group *G* on the hyperfinite type II_{1} factor *R*, up to conjugacy, associated with normal subgroups of *G*, characteristic invariants and inner invariants.

## Keywords

Normal Subgroup Finite Group Compact Group Relative Entropy Reduction Theory
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## References

- [1]V. Jones,
*Actions of finite groups on the hyperfinite type*II_{1}*factor*, Memoirs of A.M.S. 237 (1980).Google Scholar - [2]V. Jones,
*Index of sub factors*, Invention Math. 72 (1983), 1–25.MATHCrossRefGoogle Scholar - [3]S. Kawakami and H. Yoshida,
*Actions of a finite group on finite von Neumann algebras and the relative entropy*, J. Math. Soc. Japan 39 (1987), 609–626.MathSciNetMATHCrossRefGoogle Scholar - [4]S. Kawakami and H. Yoshida, Math. Japon.,
*Reduction theory on the relative entropy*, 33(1988), 975–990.MathSciNetMATHGoogle Scholar - [5]G.W. Mackey,
*Unitary representations of group extensions*I, Acta. Math. 99 (1958), 265–311.MathSciNetMATHCrossRefGoogle Scholar - [6]A. Ocneanu,
*Actions of discrete amenable groups on von Neumann algebras*, Springer Lecture Notes, 1138 (1985).Google Scholar - [7]M. Pimsner and S. Popa,
*Entropy and index for subfactors*, Ann. Sci. Éc. Norm. Sup. 19 (1986), 57–106.MathSciNetMATHGoogle Scholar

## Copyright information

© Springer Science+Business Media New York 1991