Semigroup Operator Algebras and Quantum Semigroups
Marat Aukhadiev, Suren Grigoryan, Ekaterina Lipacheva

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 -algebra is presented. This -algebra appears as a "deformation" of the continuous functions algebra on a compact abelian group. Considering semigroup -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 -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.
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Taxonomy
TopicsAdvanced Operator Algebra Research · Algebraic structures and combinatorial models · Advanced Topics in Algebra
