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A characterization of coactions whose fixed-point algebras contain special maximal abelian $\ast$-subalgebras

Published online by Cambridge University Press:  14 November 2006

HISASHI AOI  and
TAKEHIKO YAMANOUCHI
HISASHI AOI
Affiliation:
Department of Mathematics, Faculty of Science, Hokkaido University, Sapporo 060-0810, Japan (e-mail: [email protected], [email protected]) Department of Mathematics, Faculty of Science and Technology, Sophia University, Tokyo 102-8854, Japan (e-mail: [email protected]).
TAKEHIKO YAMANOUCHI
Affiliation:
Department of Mathematics, Faculty of Science, Hokkaido University, Sapporo 060-0810, Japan (e-mail: [email protected], [email protected])
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Abstract

It is shown that, for the von Neumann algebra $A$ obtained from a principal measured groupoid $\mathcal{R}$ with the diagonal subalgebra $D$ of $A$, there exists a natural ‘bijective’ correspondence between coactions on $A$ that fix $D$ pointwise and Borel 1-cocycles on $\mathcal{R}$. As an application of this result, we classify a certain type of coactions on approximately finite-dimensional type II factors up to cocycle conjugacy. By using our characterization of coactions mentioned above, we are also able to generalize to some extent those results of Zimmer concerning 1-cocycles on ergodic equivalence relations into compact groups.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 26 , Issue 6 , December 2006 , pp. 1673 - 1706
Copyright
2006 Cambridge University Press

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