Semigroup Operator Algebras and Quantum Semigroups

Abstract

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.

announced at NCGQG 2012, Oslo. To appear in Sbornik:Mathematics