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Classification of actions of discrete Kac algebras on injective factors

About this Title

Toshihiko Masuda,

Graduate School of Mathematics, Kyushu University, Fukuoka, 819-0395

, Japan

and Reiji Tomatsu,

Department of Mathematics, Hokkaido University, Hokkaido 060-0810

, Japan

Publication: Memoirs of the American Mathematical Society
Publication Year: 2017; Volume 245, Number 1160
ISBNs: 978-1-4704-2055-0 (print); 978-1-4704-3609-4 (online)
Published electronically: July 26, 2016
Keywords: von Neumann algebra, discrete Kac algebra, action
MSC: Primary 46L65; Secondary 46L55

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Table of Contents


  • Introduction
  • 1. Preliminary
  • 2. Canonical extension of irreducible endomorphisms
  • 3. Kac algebras
  • 4. Classification of modular kernels
  • 5. Classification of actions with non-trivial modular parts
  • 6. Classification of centrally free actions
  • 7. Related problems
  • 8. Appendix


We will study two kinds of actions of a discrete amenable Kac algebra. The first one is an action whose modular part is normal. We will construct a new invariant which generalizes a characteristic invariant for a discrete group action, and we will present a complete classification. The second is a centrally free action. By constructing a Rohlin tower in an asymptotic centralizer, we will show that the Connes–Takesaki module is a complete invariant.

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