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Zbl 0804.46078
Baaj, Saad; Skandalis, Georges
Multiplicative unitaries and duality for crossed products of $C\sp*$-algebras. (Unitaires multiplicatifs et dualité pour les produits croisés de $C\sp*$-algèbres.) (French)
[J] Ann. Sci. Éc. Norm. Supér. (4) 26, No. 4, 425-488 (1993). ISSN 0012-9593

Summary: Let $H$ be a Hilbert space. A unitary operator $V\in {\cal L}(H\otimes H)$ is said to be multiplicative if it satisfies the pentagone equation $V\sb{12} V\sb{13} V\sb{23}= V\sb{23} V\sb{12}$. 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 $C\sp*$- algebras. Moreover, we establish Takesaki-Takai duality results, using an adaptation of the method of [{\it M. Enock}, J. Funct. Anal. 26, 16-47 (1977; Zbl 0366.46053)].\par 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 2000:: *46L55 Noncommutative dynamical systems
47L50 Dual spaces of operator algebras
46L05 General theory of C*-algebras

Keywords: locally compact quantum group; regularity; irreducibility; crossed products; Hilbert space; unitary operator; pentagone equation; operator algebras with duality; multiplicative unitary; dual Hopf $C\sp*$- algebras; Takesaki-Takai duality; commutativity

Citations: Zbl 0366.46053

Cited in: Zbl 1091.46513 Zbl 1099.46039 Zbl 1060.46515 Zbl 1009.46037 Zbl 1003.46040 Zbl 0998.16027 Zbl 0998.46040 Zbl 0980.46048 Zbl 0876.46044 Zbl 0854.46053 Zbl 0840.22010 Zbl 0830.46064 Zbl 0839.46055 Zbl 0818.17014 Zbl 0819.46054

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