Compact Abelian group actions on injective factors (Q1187766)

The final step of classification of compact Abelian group actions on injective factors is given. Actions of a compact (separable) Abelian group on injective type III factors ae classified up to conjugacy: Two actions are conjugate if the dual invariant \(\partial(\alpha)\) and the inner invariant \(\tau(\alpha)\) defined by \textit{V. F. R. Jones} and \textit{M. Takesaki} [Acta Math. 153, 213-258 (1984; Zbl 0588.46042), Definition 3.2.1] are the same (Theorem 3.1 (ii)). A necessary and sufficient condition for two actions to be cocycle conjugate is also given (Theorem 3.1 (i)). For prime actions with properly infinite fixed point algebras, a simpler classification is given (Theorem 3.6). A detailed study of the 1-dimensional torus case is given (Theorem 4.1). Centrally ergodic actions of (countable) discrete Abelian groups on injective von Neumann algebras of type III up to cocycle conjugacy are also classified (Theorem 2.2). This result is used to derive the above classification of actions of a compact Abelian group through Takesaki duality.