Skip to main content
Log in

Global Homological Dimension of Radical Banach Algebras of Power Series

  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

We show that the global dimension of a broad class of radical Banach algebras of power series is at least 3 and obtain applications to cohomology groups.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. A. Ya. Khelemskii, Homology in Banach and Topologial Algebras (Izd.Moskov. Univ., Moscow, 1986) [in Russian].

    Google Scholar 

  2. Yu. V. Selivanov, “Homological dimensions of tensor products of Banach algebras,” in Banach Algebras’ 97 (Walter de Gruyter, Berlin, 1998), pp. 441–459.

    Google Scholar 

  3. S. B. Tabaldyev, “Additivity of homological dimensions for a class of Banach algebras,” Funktsional. Anal. Prilozhen. 40 (3), 93–95 (2006) [Functional Anal. Appl. 40 (3), 244–246 (2006)].

    Article  MathSciNet  MATH  Google Scholar 

  4. Yu.V. Selivanov, “Lower bounds for homological dimensions of Banach algebras,” Mat. Sb. 198 (9), 133–160 (2007) [Sb.Math. 198 (9), 1351–1377 (2007)].

    Article  MathSciNet  MATH  Google Scholar 

  5. A. Ya. Khelemskii, “Global dimension of a Banach function algebra is different from unity,” Funktsional. Anal. Prilozhen. 6 (2), 95–96 (1972) [Functional Anal. Appl. 6 (2), 166–168 (1972)].

    Google Scholar 

  6. A. Ya. Khelemskii, “Smallest values assumed by the global homological dimension of Banach function algebras,” Trudy Sem. Petrovsk. 3, 223–242 (1978) [Am.Math. Soc. Transl. Ser. 2 124, 75–96 (1984)].

    Google Scholar 

  7. S. Pott, “An account on the global homological dimension theorem of A. Ya. Helemskii,” Ann. Univ. Sarav. Ser. Math. 9 (3), 155–194 (1999).

    MathSciNet  MATH  Google Scholar 

  8. A. Ya. Khelemskii, “On a method for calculating and estimating the global homological dimension of Banach algebras,” Mat. Sb. 87 (129) (1), 122–135 (1972) [Math. USSR-Sb. 16 (1), 125–138 (1972)].

    MathSciNet  Google Scholar 

  9. Yu. V. Selivanov, “Biprojective Banach algebras, their structure, cohomologies, and connection with nuclear operators,” Funktsional. Anal. Prilozhen. 10 (1), 89–90 (1976) [Functional Anal. Appl. 10 (1), 78–79 (1976)].

    MATH  Google Scholar 

  10. Z. A. Lykova, “A lower estimate of the global homological dimension of infinite-dimensional CCR-algebras,” UspekhiMat. Nauk 41 (1 (247)), 197–198 (1986) [RussianMath. Surveys 41 (1), 233–234 (1986)].

    MATH  Google Scholar 

  11. O. Yu. Aristov, “The global dimension theorem for non-unital and certain other separable C*-algebras,” Mat. Sb. 186 (9), 3–18 (1995) [Sb.Math. 186 (9), 1223–1239 (1995)].

    MathSciNet  Google Scholar 

  12. Yu. V. Selivanov, “Coretraction problems and homological properties of Banach algebras,” in Topological Homology (Nova Sci. Publ., Huntington, NY, 2000), pp. 145–199.

    Google Scholar 

  13. F. Ghahramani and Yu. V. Selivanov, “The global dimension theorem for weighted convolution algebras,” Proc. Edinburgh Math. Soc. (2) 41 (2), 393–406 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  14. Yu. V. Selivanov, “Strongly non-complemented subspaces of Banach spaces and the homological dimension of Banach modules,” Uspekhi Mat. Nauk 49 (1 (295)), 223–224 (1994) [Russian Math. Surveys 49 (1), 245–246 (1994)].

    MATH  Google Scholar 

  15. H. G. Dales, Banach Algebras and Automatic Continuity, in London Math. Soc. Monogr. (N. S.) (Clarendon Press, Oxford, 2000), Vol.24.

  16. Yu. V. Selivanov, “A problem in the geometry of Banach spaces,” UspekhiMat. Nauk 48 (4 (292)), 237–238 (1993) [Russian Math. Surveys 48 (4), 251–253 (1993)].

    MathSciNet  MATH  Google Scholar 

  17. P. Wojtaszczyk, Banach Spaces for Analysts, in Cambridge Stud. Adv. Math. (Cambridge Univ. Press, Cambridge, 1991), Vol.25.

    Book  MATH  Google Scholar 

  18. A. Ya. Khelemskii, Banach and Polynormed Algebras: General Theory, Representations, and Homology (Nauka, Moscow, 1989) [in Russian].

    Google Scholar 

  19. Yu. V. Selivanov, “Computing and estimating the global dimension in certain classes of Banach algebras,” Math. Scand. 72, 85–98 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  20. I. M. Gel’fand, D. A. Raikov, and G. E. Shilov, Commutative Normed Rings (Fizmatgiz, Moscow, 1960) [in Russian].

    MATH  Google Scholar 

  21. W. G. Bade, H. G. Dales, and K. B. Laursen, “Multipliers of radical Banach algebras of power series,” in Mem. Amer. Math. Soc. (Amer.Math. Soc., Providence, RI, 1984), Vol. 49, No.303.

  22. Yu. V. Selivanov, “Classes of Banach algebras of global dimension infinity,” in Banach Algebras and Their Applications, Contemp. Math. (Amer.Math. Soc., Providence, RI, 2004), Vol. 363, pp. 321–333.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Yu. V. Selivanov.

Additional information

Original Russian Text © Yu. V. Selivanov, 2018, published in Matematicheskie Zametki, 2018, Vol. 104, No. 5, pp. 737–744.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Selivanov, Y.V. Global Homological Dimension of Radical Banach Algebras of Power Series. Math Notes 104, 720–726 (2018). https://doi.org/10.1134/S0001434618110135

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0001434618110135

Keywords

Navigation