Injectivity and Nuclearity Properties for Real C*-Algebras
- Authors: Rakhimov A.A.1, Nurillaev M.E.2, Boltaev K.K.2
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Affiliations:
- National University of Uzbekistan named after M. Ulugbek
- Tashkent State Pedagogical University named after Nizami
- Issue: Vol 67, No 4 (2021): Science — Technology — Education — Mathematics — Medicine
- Pages: 755-765
- Section: Articles
- URL: https://journals.rudn.ru/CMFD/article/view/30081
- DOI: https://doi.org/10.22363/2413-3639-2021-67-4-755-765
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Abstract
In this paper, we study injective and nuclear real W*- and C*-algebras. The connection of these concepts with similar concepts of enveloping W*- and C*-algebras is considered. The equivalence of the concepts of injectivity and nuclearity for real C*-algebras is shown. As a consequence, nuclear real factors of types II1, II∞, III1, III0 and IIIλ (0< λ<1) are completely described.
About the authors
A. A. Rakhimov
National University of Uzbekistan named after M. Ulugbek
Author for correspondence.
Email: rakhimov@ktu.edu.tr
Tashkent, Uzbekistan
M. E. Nurillaev
Tashkent State Pedagogical University named after Nizami
Email: nur11laev_muzaffar@mail.ru
Tashkent, Uzbekistan
Kh. Kh. Boltaev
Tashkent State Pedagogical University named after Nizami
Email: bkhabibzhan@mail.ru
Tashkent, Uzbekistan
References
- Аюпов Ш. А. Классификация инъективных JW-факторов// Функц. анализ и его прилож. - 1984. - 18, № 3. - C. 67-68.
- Фон Нейман Дж. Обобщение математического аппарата квантовой механики методами абстрактной алгебры. I// Мат. сб. - 1936. -1, № 4. - С. 415-485.
- Фон Нейман Дж. Избранные труды по функциональному анализу. I, II. - М.: Наука, 1987.
- Ayupov Sh. А., Rakhimov А. А., Usmanov Sh. M. Jordan, Real and Lie structures in operator algebras. - Dordrecht: Kluwer Academic Publishers, 1997.
- Ayupov Sh. A., Rakhimov A. A. Real W*-algebras, actions of groups and index theory for real factors. - Beau-Bassin: VDM Verlag Dr. Muller, 2010.¨
- Connes A. Classification of injective facteurs// Ann. Math. - 1976. -104, № 1. - С. 73-115.
- Dixmier J. Les algebres d’operateurs dans l’espace Hilbertien. - Paris: Gauthier-Villars, 1969.
- Grothendieck A. Resume des results essentiels dans la theorie des produits tensoriels topologiques et des espaces nucleaires// Ann. Inst. Fourier (Grenoble). - 1954. -4. - С. 73-112.
- Grothendieck A. Produits tensoriels topologiques et des espaces nucleaires// S´ eminaire N. Bourbaki. -´ 1954. -69. - С. 193-200.
- Li B. R. Real operator algebras. - River Edge: World Scientific, 2003.
- Murray F., von Neumann J. On rings of operators. I// Ann. Math. - 1936. -37. - С. 116-229.
- Rakhimov А. А., Nurillaev M. E. On property of injectivity for real W*-algebras and JW-algebras// Positivity. - 2018. - 22. - С. 1345-1354.
- Sakai S. C*-algebras and W*-algebras. - Berlin: Springer, 1971.
- Stacey P. J. Real structure in unital separable simple C*-algebras with tracial rank zero and with a unique tracial state// New York J. Math. - 2006. -12. - С. 269-273.
- Stormer E. Real structure in the hyperfinite factor// Duke Math. J. - 1980. -47, № 1. - С. 145-153.
- Takesaki M. Theory of operator algebras. I, II, III. - Berlin: Springer, 1979.
