Extensions of Semigroups and Morphisms of Semigroup $ C^{*} $-Algebras

Lipacheva, E. V.

doi:10.1134/s0037446621010079

Cited by 5 publications

(4 citation statements)

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“…We observe that Theorem 2.1 can be proved also without using Lemma 2.16 from [8]. This can be done in the same way how Theorem 3.1 was proved in [22].…”

Section: Preliminariesmentioning

confidence: 71%

“…In this sections we enlarge the results obtained in work [22] on embeddings of 𝐶 * -algebras corresponding to semigroups forming normal extension of semigroups.…”

Section: Preliminariesmentioning

confidence: 91%

“…This paper is aimed on finding the relations between the extensions of the semigroups and corresponding semigroup 𝐶 * -algebras and completes the studies made in works [20]- [22]. We consider one of the simplest types of normal extensions, namely, trivial extensions.…”

Section: Introductionmentioning

confidence: 99%

“…In Section 1 we recall needed facts from the theory of semigroup extensions, theory of semigroup 𝐶 * -algebras and Banach modules. Section 2 is devoted to the question on functoriality of morphisms of semigroup 𝐶 * -algebras, which raised in a general form in work [8] and was studied in [22] for reduced semigroup 𝐶 *algebras constructed by semigroups, one of which is a normal extension of another. In Section 3 on the underlaying space of the 𝐶 * -algebra 𝐶 * 𝑟 (𝐿), we introduce and study the structure of Banach 𝐶 * 𝑟 (𝑆)-module under the conditions that the semigroup 𝐿 is a trivial extension of 𝑆 by means of the group.…”

Section: Introductionmentioning

confidence: 99%

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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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“…We observe that Theorem 2.1 can be proved also without using Lemma 2.16 from [8]. This can be done in the same way how Theorem 3.1 was proved in [22].…”

Section: Preliminariesmentioning

confidence: 71%

“…In this sections we enlarge the results obtained in work [22] on embeddings of 𝐶 * -algebras corresponding to semigroups forming normal extension of semigroups.…”

Section: Preliminariesmentioning

confidence: 91%