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Aspects of compact quantum group theory. (English) Zbl 1052.46060

Summary: We show that if a compact quantum semigroup satisfies certain weak cancellation laws, then it admits a Haar measure, and using this we show that it is a compact quantum group. Thus, we obtain a new characterization of a compact quantum group. We also give a necessary and sufficient algebraic condition for the Haar measure of a compact quantum group to be faithful, in the case that its coordinate \(C^*\)-algebra is exact. A representation is given for the linear dual of the Hopf \(*\)-algebra of a compact quantum group, and a functional calculus for unbounded linear functionals is derived.

MSC:

46L89 Other “noncommutative” mathematics based on \(C^*\)-algebra theory
58B32 Geometry of quantum groups
46L65 Quantizations, deformations for selfadjoint operator algebras
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