Semigroup Operator Algebras and Quantum Semigroups

A detailed study of the semigroup $C^\ast$-algebra is presented. This $C^\ast$-algebra appears as a "deformation" of the continuous functions algebra on a compact abelian group. Considering semigroup $C^\ast$-algebras in this framework we construct a compact quantum semigroups category. Then the initial group is a compact subgroup of the new compact quantum semigroup, the natural action of this group is described. The dual space of such $C^\ast$-algebra is endowed with Banach *-algebra structure, which contains the algebra of measures on a compact group. The dense weak Hopf *-algebra is given. It is shown that the constructed category of compact quantum semigroups can be embedded to the category of abelian semigroups.