We introduce quantum semigroup compactifications and study the universal quantum semigroup compactification of a coamenable locally compact quantum group. If $G$ is a classical locally compact group, the universal semigroup compactification corresponds to the C*-algebra of the bounded left uniformly continuous functions on $G$, so we study the analogous C*-algebra associated with a locally compact quantum group.
"Quantum semigroup compactifications and uniform continuity on locally compact quantum groups." Illinois J. Math. 54 (2) 469 - 483, Summer 2010. https://doi.org/10.1215/ijm/1318598668