Quantum semigroup compactifications and uniform continuity on locally compact quantum groups
The author considers locally compact quantum groups introduced by \textit{S.~Vaes} and \textit{J.~Kustermans} [Math. Scand. 92, No. 1, 68--92 (2003; Zbl 1034.46067)]; such a group, \({\mathbb G}\), is a von Neumann algebra with an additional co-multiplication structure. Using the GNS-construction, one can define the corresponding \(C^*\)-algebra \(C_0({\mathbb G})\) by taking the norm closure of a representation of \({\mathbb G}\) on a Hilbert space. In the case \({\mathbb G}=G\) is a (classical) locally compact group, there exists a canonical compactification in terms of the \(C^*\)-subalgebras of algebra \(C_0(G)\); the compactification turns \(G\) into a semi-group. In the paper under review, the author extends the classical construction to the quantum groups using the \(C^*\)-subalgebras of the algebra \(C_0({\mathbb G})\); basic properties of such a compactification are established.
scientific article; zbMATH DE number 5981639
scientific article