We show the uniqueness of minimal actions of a compact Kac algebra with amenable dual on the AFD factor of type II 1 . This particularly implies the uniqueness of minimal actions of a compact group. Our main tools are a Rohlin type theorem, the 2-cohomology vanishing theorem, and the Evans-Kishimoto type intertwining argument.

We will introduce the Rohlin property for flows on von Neumann algebras and classify them up to strong cocycle conjugacy. This result provides alternative approaches to some preceding results such as Kawahigashi's classification of flows on the injective type II 1 factor, the classification of injective type III factors due to Connes, Krieger and Haagerup and the non-fullness of type III 0 factors. Several concrete examples are also studied.

Let G be a co-amenable compact quantum group. We show that a right coideal of G is of quotient type if and only if it is the range of a conditional expectation preserving the Haar state and is globally invariant under the left action of the dual discrete quantum group. We apply this result to theory of Poisson boundaries introduced by Izumi for discrete quantum groups and generalize a work of Izumi-Neshveyev-Tuset on SU q (N ) for coamenable compact quantum groups with the commutative fusion rules. More precisely, we prove that the Poisson integral is an isomorphism between the Poisson boundary and the right coideal of quotient type by maximal quantum subgroup of Kac type. In particular, the Poisson boundary and the quantum flag manifold are isomorphic for any q-deformed classical compact Lie group.

Z.-J. Ruan has shown that several amenability conditions are all equivalent in the case of discrete Kac algebras. In this paper, we extend this work to the case of discrete quantum groups. That is, we show that a discrete quantum group, where we do not assume its unimodularity, has an invariant mean if and only if it is strongly Voiculescu amenable.2000 Mathematics Subject Classification. 46L65.

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