Endomorphisms of the Toeplitz algebra
This article describes all injective endomorphisms of the classical Toeplitz algebra. Their connection with endomorphisms of the algebra of continuous functions on the unit circle and with coverings over the unit circle was considered. It was shown that each non-unitary isometry V in the Toeplitz al...
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| Format: | Article |
| Language: | English |
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Kazan Federal University
2023-11-01
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| Series: | Учёные записки Казанского университета: Серия Физико-математические науки |
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| Online Access: | https://uzakufismat.elpub.ru/jour/article/view/5 |
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| author | S. A. Grigoryan A. Yu. Kuznetsova |
| author_facet | S. A. Grigoryan A. Yu. Kuznetsova |
| author_sort | S. A. Grigoryan |
| collection | DOAJ |
| description | This article describes all injective endomorphisms of the classical Toeplitz algebra. Their connection with endomorphisms of the algebra of continuous functions on the unit circle and with coverings over the unit circle was considered. It was shown that each non-unitary isometry V in the Toeplitz algebra determines the identity preserving endomorphism, as well as the class of its compact perturbations, i.e., identity non-preserving endomorphisms, defined by partial isometries {V P}, where P is a projection of finite codimension. The notions of T -equivalence of endomorphisms and T -equivalence up to a compact perturbation were introduced. An example was provided wherein the isometries are unitarily equivalent but the corresponding endomorphisms fall into different equivalence classes. Of all endomorphisms, the ones belonging to the class of Blaschke endomorphisms, which are analogous to endomorphisms of the discalgebra and generate unbranched coverings over the unit circle, were singled out. |
| format | Article |
| id | doaj-art-f5f327ed36a24b7b90eb9c7efbf13cb4 |
| institution | Kabale University |
| issn | 2541-7746 2500-2198 |
| language | English |
| publishDate | 2023-11-01 |
| publisher | Kazan Federal University |
| record_format | Article |
| series | Учёные записки Казанского университета: Серия Физико-математические науки |
| spelling | doaj-art-f5f327ed36a24b7b90eb9c7efbf13cb42025-02-02T23:06:08ZengKazan Federal UniversityУчёные записки Казанского университета: Серия Физико-математические науки2541-77462500-21982023-11-011651354810.26907/2541-7746.2023.1.35-484Endomorphisms of the Toeplitz algebraS. A. Grigoryan0A. Yu. Kuznetsova1Kazan State Power Engineering UniversityKazan Federal UniversityThis article describes all injective endomorphisms of the classical Toeplitz algebra. Their connection with endomorphisms of the algebra of continuous functions on the unit circle and with coverings over the unit circle was considered. It was shown that each non-unitary isometry V in the Toeplitz algebra determines the identity preserving endomorphism, as well as the class of its compact perturbations, i.e., identity non-preserving endomorphisms, defined by partial isometries {V P}, where P is a projection of finite codimension. The notions of T -equivalence of endomorphisms and T -equivalence up to a compact perturbation were introduced. An example was provided wherein the isometries are unitarily equivalent but the corresponding endomorphisms fall into different equivalence classes. Of all endomorphisms, the ones belonging to the class of Blaschke endomorphisms, which are analogous to endomorphisms of the discalgebra and generate unbranched coverings over the unit circle, were singled out.https://uzakufismat.elpub.ru/jour/article/view/5c<sup>∗</sup> -алгебраtoeplitz algebraendomorphismautomorphismfinite blaschke productfredholm operatorpartial isometry |
| spellingShingle | S. A. Grigoryan A. Yu. Kuznetsova Endomorphisms of the Toeplitz algebra Учёные записки Казанского университета: Серия Физико-математические науки c<sup>∗</sup> -алгебра toeplitz algebra endomorphism automorphism finite blaschke product fredholm operator partial isometry |
| title | Endomorphisms of the Toeplitz algebra |
| title_full | Endomorphisms of the Toeplitz algebra |
| title_fullStr | Endomorphisms of the Toeplitz algebra |
| title_full_unstemmed | Endomorphisms of the Toeplitz algebra |
| title_short | Endomorphisms of the Toeplitz algebra |
| title_sort | endomorphisms of the toeplitz algebra |
| topic | c<sup>∗</sup> -алгебра toeplitz algebra endomorphism automorphism finite blaschke product fredholm operator partial isometry |
| url | https://uzakufismat.elpub.ru/jour/article/view/5 |
| work_keys_str_mv | AT sagrigoryan endomorphismsofthetoeplitzalgebra AT ayukuznetsova endomorphismsofthetoeplitzalgebra |