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Journal of Operator Theory

Volume 41, Issue 2, Spring 1999  pp. 261-289.

Compact quantum hypergroups

Authors:  Yu.A. Chapovsky (1), and L.I. Vainerman (2)
Author institution: (1) Institute of Mathematics, Ukrainian National Academy of Sciences, ul. Tereshchinkivs'ka, 3, Kiev 262601, Ukraine
(2) International Solomon University, Zabolotny St., 38, apt. 61, Kiev 252187, Ukraine

Summary:  A compact quantum hypergroup is a unital $C^*$-algebra equipped with a completely positive coassociative coproduct. The most important examples of such a structure are associated with double cosets of compact matrix pseudogroups in the sense of S.L.~Woronowicz. We give a precise definition of a compact quantum hypergroup; prove existence and uniqueness of the Haar measure, establish orthogonality relations for matrix elements of irreducible corepresentations; construct a Peter-Weyl theory for irreducible corepresentations.

Keywords:  $C^*$-algebra, coproduct, Haar measure, hypergroup, quantum group, quantum homogeneous space, representation

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