vestifmИзвестия Национальной академии наук Беларуси. Серия физико-математических наукProceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series1561-24302524-2415The Republican Unitary Enterprise Publishing House "Belaruskaya Navuka"10.29235/1561-2430-2021-57-2-185-189vestifm-584Research ArticleМАТЕМАТИКАMATHEMATICSn-Однородные C*-алгебрыn-Homogeneous C*-algebrasЩукинМ. В.ShchukinM. V.

Щукин Михаил Владимирович – кандидат физико-математических наук, доцент кафедры «Высшая математика»

ул. Хмельницкого, 9, 220013, г. Минск, Республика Беларусь

Mikhail V. Shchukin – Ph. D. (Physics and Mathematics), Assistant Professor of the Department “Higher Mathematics”

9, Khmel’nitskii Str., 220013, Minsk, Republic of Belarus

mvshchukin@bntu.byБелорусский национальный технический университетBelarusian National Technical University202115072021572185189Copyright © Щукин М.В., 20212021Щукин М.В.Shchukin M.V.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://vestifm.belnauka.by/jour/article/view/584

Рассматриваются результаты, касающиеся n-однородных С*-алгебр. Приводятся классические результаты Ж. Фелла, Ж. Томияма, М. Такесаки, описывающие n-однородную С*-алгебру как алгебру всех непрерывных сечений соответствующего алгебраического расслоения. Посредством этой геометрической интерпретации, различные авторы описывали классы n-однородных С*-алгебр с пространством примитивных идеалов, гомеоморфным двумерной сфере S2, трехмерной сфере S3, двумерному тору T2, трехмерному тору T3, произвольному связному ориентируемому и неориентируемому компактному двумерному многообразию. Также А. Антоневич и Н. Крупник задавали различные структуры на множестве классов эквивалентности алгебраических расслоений на сферах. Дальнейшая работа в этом направлении может состоять в описании классов эквивалентности алгебраических расслоений над трехмерными, четырехмерными многообразиями и т. д.

The classical results by J. Fell, J. Tomiyama, M. Takesaki describe n-homogeneous С*-algebras as algebras of all continuous sections for an appropriate algebraic bundle. By using this realization, several authors described the set of n-homogeneous С*-algebras with different spaces of primitive ideals. In 1974 F. Krauss and T. Lawson described the set of all n-homogeneous С*-algebras whose space Prim of primitive ideals is homeomorphic to the sphere S2. Suppose the space PrimA of primitive ideals is homeomorphic to the sphere S3 for some n-homogeneous С*-algebra A. In this case, these authors showed that the algebra A is isomorphic to the algebra C(S3,Cn×n). If n ≥ 2 then there are countably many pairwise non-isomorphic n-homogeneous С*-algebras A such that PrimA ≅ S 4. Further, let n ≥ 3. There is only one n-homogeneous С*-algebra A such that PrimA ≅ S 5. There are two non-isomorphic 2-homogeneous С*-algebras A and B with space PrimA ≅ S 5. On the other hand, algebraic bundles over the torus T 2 are described by a residue class [p] in Z/nZ = π1(PUn). Two such bundles with classes [pi] produce isomorphic С*-algebras if and only if [p1] = ±[p2]. An algebraic bundle over the torus T 3 is determined by three residue classes in Z/nZ. Anatolii Antonevich and Nahum Krupnik introduced some structures on the set of algebraic bundles over the sphere S2. Algebraic bundles over the compact connected two-dimensional oriented manifolds were considered by the author. In this case, the set of non-equivalent algebraic bundles over such space is like the set of algebraic bundles over the torus T2. Further advances could be in describing the set of algebraic bundles over the 3-dimensional manifolds.

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The authors declare that there are no conflicts of interest present.