Semigroup Operator Algebras and Quantum Semigroups

TL;DR

This paper explores the structure of semigroup $C^*$-algebras, their relation to quantum semigroups, and constructs a category of compact quantum semigroups with connections to classical groups and measure algebras.

Contribution

It introduces a new framework linking semigroup $C^*$-algebras with quantum semigroups and constructs an embedding into abelian semigroup categories.

Findings

Construction of a compact quantum semigroups category

Embedding of the quantum semigroups category into abelian semigroups

Identification of the dual space with measure algebras on compact groups

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.