Injectivity and Nuclearity Properties for Real C*-Algebras

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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.
Tashkent, Uzbekistan

M. E. Nurillaev

Tashkent State Pedagogical University named after Nizami

Tashkent, Uzbekistan

Kh. Kh. Boltaev

Tashkent State Pedagogical University named after Nizami

Tashkent, Uzbekistan


  1. Аюпов Ш. А. Классификация инъективных JW-факторов// Функц. анализ и его прилож. - 1984. - 18, № 3. - C. 67-68.
  2. Фон Нейман Дж. Обобщение математического аппарата квантовой механики методами абстрактной алгебры. I// Мат. сб. - 1936. -1, № 4. - С. 415-485.
  3. Фон Нейман Дж. Избранные труды по функциональному анализу. I, II. - М.: Наука, 1987.
  4. Ayupov Sh. А., Rakhimov А. А., Usmanov Sh. M. Jordan, Real and Lie structures in operator algebras. - Dordrecht: Kluwer Academic Publishers, 1997.
  5. 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.¨
  6. Connes A. Classification of injective facteurs// Ann. Math. - 1976. -104, № 1. - С. 73-115.
  7. Dixmier J. Les algebres d’operateurs dans l’espace Hilbertien. - Paris: Gauthier-Villars, 1969.
  8. 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.
  9. Grothendieck A. Produits tensoriels topologiques et des espaces nucleaires// S´ eminaire N. Bourbaki. -´ 1954. -69. - С. 193-200.
  10. Li B. R. Real operator algebras. - River Edge: World Scientific, 2003.
  11. Murray F., von Neumann J. On rings of operators. I// Ann. Math. - 1936. -37. - С. 116-229.
  12. Rakhimov А. А., Nurillaev M. E. On property of injectivity for real W*-algebras and JW-algebras// Positivity. - 2018. - 22. - С. 1345-1354.
  13. Sakai S. C*-algebras and W*-algebras. - Berlin: Springer, 1971.
  14. 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.
  15. Stormer E. Real structure in the hyperfinite factor// Duke Math. J. - 1980. -47, № 1. - С. 145-153.
  16. Takesaki M. Theory of operator algebras. I, II, III. - Berlin: Springer, 1979.

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