Irreducible inclusions of factors, multiplicative unitaries, and Kac algebras. (English) Zbl 0847.22003

From an irreducible depth 2 inclusion of factors, verifying a regularity condition, the authors construct a multiplicative unitary, and an action, at every level of the canonical tower constructed from the inclusion; when this inclusion admits a faithful semifinite normal operator-valued weight, stronger conditions are given, and the tower appears then as a crossed-product construction. In particular they rederive Herman and Ocneanu’s results when the inclusion admits a faithful normal conditional expectation, and the tower is then the crossed-product construction, alternatively by a compact quantum group and by its dual, and, more precisely, according to Yamagami’s result, by a compact type Kac algebra and by its dual.


22D10 Unitary representations of locally compact groups
22D12 Other representations of locally compact groups
22D25 \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations
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