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Marat Aukhadiev

Publications and source records attributed to Marat Aukhadiev.

4 recordsLinked to original sources
arXiv2016

An inverse semigroup approach to the C*-algebras and crossed products of cancellative semigroups

We give a new definition of the semigroup C*-algebra of a left cancellative semigroup, which resolves problems of the construction by X. Li. Namely, the new construction is functorial, and the independence of ideals in the semigroup does not influence the independence of the generators. It has a group C*-algebra as a natural quotient. The C*-algebra of the old construction is a quotient of the new one. All this applies both to the full and reduced C*-algebras. The construction is based on the universal inverse semigroup generated by a left cancellative semigroup. We apply this approach to connect amenability of a semigroup to nuclearity of its C*-algebra. Large classes of actions of these semigroups are in one-to-one correspondence, and the crossed products are isomorphic. A crossed product of a left Ore semigroup is isomorphic to the partial crossed product of the generated group.

math.OA↗
arXiv2016

Groupoids viewed as inverse semigroups

A simple observation, showing that every groupoid becomes an inverse semigroup after adding one element. In such inverse semigroups all idempotents are mutually orthogonal. This fact implies that every C*-algebra of a discrete groupoid is a C*-algebra of an inverse semigroup.

math.OA↗
arXiv2013

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.

math.OA↗