Compact quantum groups and group duality (Q1194278)

The author studies compact quantum groups (roughly, Hopf \(C^*\)-algebras with unity) and shows that if the underlying \(C^*\)-algebra of a compact quantum group is commutative, then it is isomorphic to the algebra of functions on a compact group (a consequence of the Gelfand-Naimark theorem). He also gives versions of the preceding for compact quantum semigroups and locally compact quantum groups (roughly, Hopf \(C^*\)- algebras without unity, here multipliers are used). The topological dual of the \(C^*\)-algebra of a compact quantum group \(G\) is shown to be a Banach algebra, and the finite dimensional representations of \(G\) are representations of this dual algebra. [See also \textit{S. L. Woronowicz}, Commun. Math. Phys. 111, 613-665 (1987; Zbl 0627.58034)].