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Multiplicative unitaries and duality for crossed products of \(C^*\)-algebras. (Unitaires multiplicatifs et dualité pour les produits croisés de \(C^*\)-algèbres.) (French) Zbl 0804.46078

Summary: Let \(H\) be a Hilbert space. A unitary operator \(V\in {\mathcal L}(H\otimes H)\) is said to be multiplicative if it satisfies the pentagone equation \(V_{12} V_{13} V_{23}= V_{23} V_{12}\). In many papers concerned on operator algebras with duality, a multiplicative unitary plays a fundamental role. In this paper we look for additional conditions that a multiplicative unitary should satisfy in order to correspond to a “locally compact quantum group”. We introduce two conditions: “regularity” and “irreducibility”. To any multiplicative unitary satisfying these conditions we associate two pairwise dual Hopf \(C^*\)- algebras. Moreover, we establish Takesaki-Takai duality results, using an adaptation of the method of [M. Enock, J. Funct. Anal. 26, 16-47 (1977; Zbl 0366.46053)].
If the Hilbert space is finite-dimensional or if the unitary V satisfies a commutativity condition, regularity and irreducibility are automatic. If the unitary V is of compact or discrete type, its regularity implies its irreducibility.

MSC:

46L55 Noncommutative dynamical systems
47L50 Dual spaces of operator algebras
46L05 General theory of \(C^*\)-algebras

Citations:

Zbl 0366.46053
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References:

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