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On Vector-Valued Banach Function Algebras

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Abstract

We consider vector-valued Banach function algebras on a compact Hausdorff space. Then, we define the subalgebras generated by vector-valued polynomials and rational functions, and determine their maximal ideal spaces and Šilov boundaries. We finally make use the results for a certain category of these algebras such as vector-valued Lipschitz algebras, vector-valued Dales–Davie algebras (algebras of vector-valued differentiable functions) and the algebras of vector-valued differentiable Lipschitz functions.

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References

  1. Abtahi, M.: Vector-valued characters on vector-valued function algebras. Banach J. Math. Anal. 10(3), 608–620 (2016)

    Article  MathSciNet  Google Scholar 

  2. Bade, W.G., Curtis Jr., P.C., Dales, H.G.: Amenability and weak amenability for Beurling and Lipschitz algebras. Proc. Lond. Math. Soc. 55(3), 359–377 (1987)

    Article  MathSciNet  Google Scholar 

  3. Bland, W.J., Feinstein, J.F.: Completions of normed algebras of differentiable functions. Stud. Math. 170, 89–111 (2005)

    Article  MathSciNet  Google Scholar 

  4. Cao, H.X., Zhang, J.H., Xu, Z.B.: Characterizations and extensions of Lipschitz-\(\alpha \) operators. Acta Math. Sin. Engl. Ser. 22(3), 671–678 (2006)

    Article  MathSciNet  Google Scholar 

  5. Dales, H.G.: Banach algebras and automatic continuity, London Math. Soc. Monographs 24. The Clarendon Press, Oxford (2000)

    Google Scholar 

  6. Dales, H.G., Davie, A.M.: Quasianalytic Banach function algebras. J. Funct. Anal. 13, 28–50 (1973)

    Article  MathSciNet  Google Scholar 

  7. Dales, H.G., Feinstein, J.F.: Normed algebras of differentiable functions on compact plane sets. Indian J. Pure Appl. Math. 41, 153–187 (2010)

    Article  MathSciNet  Google Scholar 

  8. de Leeuw, K.: Banach spaces of Lipschitz functions, Studia Math. 21, 55–66 (1961/1962)

  9. Esmaeili, K.: Weighted composition operators on vector-valued Lipschitz function spaces and on Zygmund type spaces. Ph.D. Thesis, Kharazmi University, Tehran, Iran (2013)

  10. Esmaeili, K., Mahyar, H.: Weighted composition operators between vector-valued Lipschitz function spaces. Banach J. Math. Anal. 7(1), 59–72 (2013)

    Article  MathSciNet  Google Scholar 

  11. Esmaeili, K., Mahyar, H.: The character spaces and the Šilov boundaries of vector-valued Lipschitz function algebras. Indian J. Pure Appl. Math. 45(6), 977–988 (2014)

    Article  MathSciNet  Google Scholar 

  12. Gamelin, T.W.: Uniform Algebras. Chelsea Publishing Company, New York (1984)

    MATH  Google Scholar 

  13. Hatori, O., Oi, S., Takagi, H.: Peculiar homomorphisms on algebras of vector valued maps. Stud. Math. 242(2), 141–163 (2018)

    Article  MathSciNet  Google Scholar 

  14. Hausner, A.: Ideals in a certain Banach algebra. Proc. Am. Math. Soc. 8, 246–249 (1957)

    Article  MathSciNet  Google Scholar 

  15. Honary, T.G.: Relations between Banach function algebras and their uniform closures. Proc. Am. Math. Soc. 109(2), 337–342 (1990)

    Article  MathSciNet  Google Scholar 

  16. Honary, T.G., Mahyar, H.: Approximation in Lipschitz algebras of infinitely differentiable functions. Bull. Korean Math. Soc. 36, 629–636 (1999)

    MathSciNet  MATH  Google Scholar 

  17. Honary, T.G., Mahyar, H.: Approximation in Lipschitz algebras. J. Quaest. Math. 23, 13–19 (2000)

    Article  MathSciNet  Google Scholar 

  18. Jarosz, K.: \({{\rm Lip}}_{Hol}(X,\alpha )\). Proc. Am. Math. Soc. 125, 3129–3130 (1997)

    Article  MathSciNet  Google Scholar 

  19. Johnson, J.A.: Banach spaces of Lipschitz functions and vector-valued Lipschitz functions. Trans. Am. Math. Soc. 148, 147–169 (1970)

    Article  MathSciNet  Google Scholar 

  20. Mahyar, H. Approximation in Lipschitz algebras and their maximal ideal space. Ph.D. Thesis, University For Teacher Education, Tehran, Iran (1994)

  21. Mahyar, H.: Compact endomorphisms of infinitely differentiable Lipschitz algebras. Rocky Mt. J. Math. 39, 193–217 (2009)

    Article  MathSciNet  Google Scholar 

  22. Nikou, A., O’Farrell: Banach algebras of vector-valued functions. Glasgow Math. J. 56(2), 419–426 (2014)

    Article  MathSciNet  Google Scholar 

  23. Sherbert, D.R.: Banach algebras of Lipschitz functions. Pac. J. Math. 13, 1387–1399 (1963)

    Article  MathSciNet  Google Scholar 

  24. Sherbert, D.R.: The structure of ideals and point derivations in Banach algebras of Lipschitz functions. Trans. Am. Math. Soc. 111, 240–272 (1964)

    Article  MathSciNet  Google Scholar 

  25. Weaver, N.: Lipschitz Algebras. World Scientific Publishing Co. Inc, River Edge (1999)

    Book  Google Scholar 

  26. Żelazko, W.: Banach Agebras. Elsevier Publishing Company, New York (1973)

    Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Mathematics, Kharazmi University, 50, Taleghani Ave., Tehran, 15618, Iran

    Hakimeh Mahyar

  2. Faculty of Engineering, Ardakan University, Ardakan, Yazd, 89518-95491, Iran

    Kobra Esmaeili

Authors
  1. Hakimeh Mahyar
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  2. Kobra Esmaeili
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Correspondence to Hakimeh Mahyar.

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Communicated by Mohammad B. Asadi.

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Mahyar, H., Esmaeili, K. On Vector-Valued Banach Function Algebras. Bull. Iran. Math. Soc. 48, 111–125 (2022). https://doi.org/10.1007/s41980-020-00504-4

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  • DOI: https://doi.org/10.1007/s41980-020-00504-4

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