On compact quantum semigroup $QS_\mathrm{red}$

We study some properties of a reduced semigroup $C^*$-algebra of a semigroup $S$. For the semigroup $C^*$-algebra generated by the deformation of the algebra of continuous functions on a compact abelian group we obtain a structure of a compact quantum semigroup. We also consider morphisms of constructed compact quantum semigroups.