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The unitary implementation of a locally compact quantum group action. (English) Zbl 1011.46058

In the present paper, the author studies actions of locally compact quantum groups on von Neumann algebras. He proves that every action has a canonical unitary implementation. This result is then used to study subfactors and inclusions of von Neumann algebras. The author gives necessary and sufficient conditions under which the inclusion is a basic construction. It is proved that the inclusion of the fixed point algebra into the von Neumann algebra is irreducible of depth 2 and regular provided the action of the quantum group is outer and integrable. The author also proves the equivalence of minimal and outer actions and generalizes the basic result that every integrable outer action with infinite fixed point algebra is a dual action.

MSC:

46L89 Other “noncommutative” mathematics based on \(C^*\)-algebra theory
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
46L37 Subfactors and their classification
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