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A $$C^ *$$-algebraic framework for quantum groups. (English) Zbl 1053.46050
This paper begins with a very readable overview of what a locally compact quantum group should be. While we know what a compact quantum group is, beyond saying that locally compact quantum groups are generalizations of locally compact groups, the noncompact case is still troublesome, and the authors explain why. They then introduce an axiom system, noting that it is not completely satisfactory, principally due to their having to introduce the Haar weight axiomatically. Their system is based on the notion of a weighted Hopf $$C^*$$-algebra. The ingredients of this central concept are a $$C^*$$-algebra $$A$$ and a coproduct $$\delta$$ forming a $$C^*$$-bialgebra ($$\delta$$ is a homomorphism from $$A$$ into the multiplier algebra of the minimal tensor product $$A\otimes A$$), a Haar weight $$h$$ and an antipode $$\kappa=R\circ \tau_{i/2}$$: $$R$$ is an involutive anti-automorphism and $$\tau(\mathbb{R})$$ is the scaling group of automorphisms of $$A$$. The authors analyse the properties of weighted Hopf $$C^*$$-algebras in detail. The results, such as the uniqueness of the Haar weight and antipode, and duality, lends support to this set of axioms as a good working system. Moreover, the paper is well written and has useful supporting appendices.

##### MSC:
 46L89 Other “noncommutative” mathematics based on $$C^*$$-algebra theory 46L65 Quantizations, deformations for selfadjoint operator algebras 20G42 Quantum groups (quantized function algebras) and their representations 22D35 Duality theorems for locally compact groups 17B37 Quantum groups (quantized enveloping algebras) and related deformations 81R50 Quantum groups and related algebraic methods applied to problems in quantum theory 46-02 Research exposition (monographs, survey articles) pertaining to functional analysis
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