On the Stone–Čech Compactification Functor and the Normal Extensions of Monoids

Berdnikov, I. S.; Gumerov, R. N.; Lipacheva, E. V.

doi:10.1134/s199508022110005x

Cited by 2 publications

(1 citation statement)

References 12 publications

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“…In studies on semigroups, various types of extensions are considered: ideal extensions, [15], Schreier extensions [16], normal extensions [17], [18]. In [19] there was studied the action of the functor of Stone-Čech compactification on normal extensions of semigroups.…”

Section: Introductionmentioning

confidence: 99%

Trivial extensions of semigroups and semigroup $C^*$-algebras

Lipacheva

2022

Ufimsk. Mat. Zh.

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The object of the study in the paper is reduced semigroup 𝐶 * -algebras for left cancellative semigroups. Such algebras are a very natural object because it is generated by isometric shift operators belonging to the image of the left regular representation of a left cancellative semigroup. These operators act in the Hilbert space consisting of all square summable complex-valued functions defined on a semigroup. We study the question on functoriality of involutive homomorphisms of semigroup 𝐶 * -algebras, that is, the existence of the canonical embedding of semigroup 𝐶 * -algebras induced by an embedding of corresponding semigroups. In order to do this, we investigate the reduced semigroup 𝐶 * -algebras associated with semigroups involved in constructing normal extensions of semigroups by groups. At the same time, in the paper we consider one of the simplest classes of extensions, namely, the class of so-called trivial extensions. It is shown that if a semigroup 𝐿 is a trivial extension of the semigroup 𝑆 by means of a group 𝐺, then there exists the embedding of the reduced semigroup 𝐶 * -algebra 𝐶 * 𝑟 (𝑆) into the 𝐶 * -algebra 𝐶 * 𝑟 (𝐿) which is induced by an embedding of the semigroup 𝑆 into the semigroup 𝐿.In the work we also introduce and study the structure of a Banach 𝐶 * 𝑟 (𝑆)-module on the underlying space of the reduced semigroup 𝐶 * -algebra 𝐶 * 𝑟 (𝐿). To do this, we use a topological grading for the 𝐶 * -algebra 𝐶 * 𝑟 (𝐿) over the group 𝐺. In the case when a semigroup 𝐿 is a trivial extension of a semigroup 𝑆 by means of a finite group, we prove the existence of the structure of a free Banach module over the reduced semigroup 𝐶 * -algebra 𝐶 * 𝑟 (𝑆) on the underlying Banach space of the semigroup 𝐶 * -algebra 𝐶 * 𝑟 (𝐿). We give examples of extensions of semigroups and reduced semigroup 𝐶 * -algebras for a more complete characterization of the issues under consideration and for revealing connections with previous results.

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Section: Introductionmentioning

confidence: 99%

Trivial extensions of semigroups and semigroup $C^*$-algebras

Lipacheva

2022

Ufimsk. Mat. Zh.

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Extensions of semigroups by the dihedral groups and semigroup C∗-algebras

Lipacheva

2022

J. Algebra Appl.

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We consider the semidirect product [Formula: see text] of the additive group [Formula: see text] of all integers and the multiplicative semigroup [Formula: see text] of integers without zero relative to a semigroup homomorphism [Formula: see text] from [Formula: see text] to the endomorphism semigroup of [Formula: see text]. It is shown that this semidirect product is a normal extension of the semigroup [Formula: see text] by the dihedral group, where [Formula: see text] is the multiplicative semigroup of all natural numbers. Further, we study the structure of [Formula: see text]-algebras associated with this extension. In particular, we prove that the reduced semigroup [Formula: see text]-algebra of the semigroup [Formula: see text] is topologically graded over the dihedral group. As a consequence, there exists a structure of a free Banach module over the reduced semigroup [Formula: see text]-algebra of [Formula: see text] in the underlying Banach space of the reduced semigroup [Formula: see text]-algebra of [Formula: see text].

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On the Stone–Čech Compactification Functor and the Normal Extensions of Monoids

Cited by 2 publications

References 12 publications

Trivial extensions of semigroups and semigroup $C^*$-algebras

Trivial extensions of semigroups and semigroup $C^*$-algebras

Extensions of semigroups by the dihedral groups and semigroup C∗-algebras

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