Operator approach to quantization of semigroups

The paper is devoted to the construction of compact quantum semigroups from semigroup $C^*$-algebras generated by the ‘deformation’ of algebras of continuous functions on compact Abelian groups. The dual space of such a $C^*$-algebra is endowed with the structure of a Banach *-algebra containing the algebra of measures on a compact group. We construct a weak Hopf *-algebra that is dense in such a compact quantum semigroup. We show that there exists an injective functor from the constructed category of compact quantum semigroups into the category of Abelian semigroups.
Bibliography: 25 titles.