SpringerLink
Log in

Diameter two properties in some vector-valued function spaces

  • Original Paper
  • Published:
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

We introduce a vector-valued version of a uniform algebra, called the vector-valued function space over a uniform algebra. The diameter two properties of the vector-valued function space over a uniform algebra on an infinite compact Hausdorff space are investigated. Every nonempty relatively weakly open subset of the unit ball of a vector-valued function space \(A(K, (X, \tau ))\) over an infinite dimensional uniform algebra has diameter two, where \(\tau \) is a locally convex Hausdorff topology on a Banach space X compatible to a dual pair. Under the assumption of X equipped with the norm topology being uniformly convex and the additional condition that \(A\otimes X\subset A(K, X)\), it is shown that Daugavet points and \(\Delta \)-points on A(K, X) over a uniform algebra A are the same, and they are characterized by the norm-attainment at a limit point of the Shilov boundary of A. In addition, a sufficient condition for the convex diametral local diameter two property of A(K, X) is also provided. Similar results also hold for an infinite dimensional uniform algebra.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abrahamsen, T., Haller, R., Lima, V., Pirk, K.: Delta- and Daugavet points in Banach spaces. Proc. Edinb. Math. Soc. 63(2), 475–496 (2020)

    Article  MathSciNet  Google Scholar 

  2. Abrahamsen, T., Lima, V., Nygaard, O.: Remarks on diameter \(2\) properties. J. Convex Anal. 20(2), 439–452 (2013)

    MathSciNet  MATH  Google Scholar 

  3. Abrahamsen, T., Nygaard, O., Põldvere, M.: New applications of extremely regular function spaces. Pac. J. Math. 301(2), 385–394 (2019)

    Article  MathSciNet  Google Scholar 

  4. Acosta, M.D., Kamińska, A.: Weak neighborhoods and the Daugavet property of the interpolation spaces \(L^1 + L^\infty \) and \(L^1\cap L^\infty \). Indiana Univ. Math. J. 57(1), 77–96 (2008)

    Article  MathSciNet  Google Scholar 

  5. Becerra Guerrero, J., López Pérez, G.: Relatively weakly open subsets of the unit ball in function spaces. J. Math. Anal. Appl. 315(2), 544–554 (2006)

    Article  MathSciNet  Google Scholar 

  6. Becerra Guerrero, J., López Pérez, G., Rueda Zoca, A.: Diametral diameter two properties in Banach spaces. J. Convex Anal. 25(3), 817–840 (2018)

    MathSciNet  MATH  Google Scholar 

  7. Choi, Y.S., García, D., Kim, S.K., Maestre, M.: Some geometric properties of disk algebras. J. Math. Anal. Appl. 409(1), 147–157 (2014)

    Article  MathSciNet  Google Scholar 

  8. Cascales, B., Guirao, A., Kadets, V.: A Bishop–Phelps–Bollobás type theorem for uniform algebras. Adv. Math. 240, 370–382 (2013)

    Article  MathSciNet  Google Scholar 

  9. Dales, H.G.: Banach Algebras and Automatic Continuity. London Mathematical Society Monographs. New Series, vol. 24. The Clarendon Press, Oxford University Press, New York (2000)

    MATH  Google Scholar 

  10. Dantas, S., Martínez-Cervantes, G., Rodríguez Abellán, J.D., Rueda Zoca, A.: Octahedral norms in free Banach lattices. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 115(1) (2021). https://doi.org/10.1007/s13398-020-00940-1

  11. Daugavet, I.K.: A property of completely continuous operators in the space \(C\). Uspekhi Mat. Nauk 18(5(113)), 157–158 (1963). (Russian)

  12. Foia̧, C., Singer, I.: Points of Diffusion of Linear Operators and Almost Diffuse Operators in Spaces of Continuous Functions. Math. Z. 87, 434–450 (1965)

  13. Ghoussoub, N., Godefroy, G., Maurey, B., Schachermayer, W.: Some topological and geometrical structures in Banach spaces. Mem. Am. Math. Soc. 70(378) (1987)

  14. Haller, R., Langemets, J., Lima, V., Nadel, R.: Symmetric strong diameter two property. Mediterr. J. Math. 16(2), Art. 35 (2019)

  15. Ivakhno, Y., Kadets, V.: Unconditional sums of spaces with bad projections. Visn. Khark. Univ. Ser. Mat. Pryki. Mat. Mekh. 645, 30–35 (2004)

    MATH  Google Scholar 

  16. Kadets, V.: The diametral strong diameter 2 property of Banach spaces is the same as the Daugavet property. Proc. Am. Math. Soc. 149, 2579–2582 (2021)

    Article  MathSciNet  Google Scholar 

  17. Kim, S.K., Lee, H.J.: A Urysohn-type theorem and the Bishop–Phelps–Bollobás theorem for holomorphic functions. J. Math. Anal. Appl. 480(2), Art. 123393 (2019)

  18. Langemets, J., Lima, V., Rueda Zoca, A.: Almost square and octahedral norms in tensor products of Banach spaces. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 111(3), 841–853 (2017)

    Article  MathSciNet  Google Scholar 

  19. Langemets, J., Pirk, K.: Stability of diametral diameter two properties. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 115(2) (2021). https://doi.org/10.1007/s13398-021-01038-y

  20. Lee, H.J.: Generalized numerical index of function algebras. J. Funct. Spaces 2019 (2019). https://doi.org/10.1155/2019/9080867

  21. Leibowitz, G.M.: Lectures on Complex Function Algebras. Scott, Foresman and Company, Northbrook (1970)

    MATH  Google Scholar 

  22. López-Pérez, G., Rueda Zoca, A.: Diameter two properties and polyhedrality. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 113, 131–135 (2019)

    Article  MathSciNet  Google Scholar 

  23. Lozanovskii, G.. Ya..: On almost integral operators in KB-spaces. Vestnik Leningr. Univ. Mat. Mekh. Astr. 21, 35–44 (1966). (Russian)

    MathSciNet  Google Scholar 

  24. Narici, L., Beckenstein, E.: Topological Vector Spaces. Pure and Applied Mathematics, 2nd edn. CRC Press, New York (2011)

    MATH  Google Scholar 

  25. Nygaard, O., Werner, D.: Slices in the unit ball of a uniform algebra. Arch. Math. (Basel) 76(6), 441–444 (2001)

    Article  MathSciNet  Google Scholar 

  26. Ostrak, A.: On the duality of the symmetric strong diameter two property in Lipschitz spaces. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 115(2) (2021). https://doi.org/10.1007/s13398-021-01018-2

  27. Pirk, K.: Diametral diameter two properties, Daugavet-, and \(\Delta \)-points in Banach spaces. PhD dissertation. Dissertationes Mathematicae Universitatis Tartuensis, 133 (2020)

  28. Rudin, W.: Functional Analysis, 2nd edn. McGraw-Hill Inc., New York (1991)

    MATH  Google Scholar 

  29. Rueda Zoca, A.: Almost squareness and strong diameter two property in tensor product spaces. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 114(2) (2021). https://doi.org/10.1007/s13398-020-00816-4

  30. Schaefer, H., Wolff, M.: Topological Vector Spaces. Graduate Texts in Mathematics, vol. 3, 2nd edn. Springer, New York (1999)

    Book  Google Scholar 

  31. Werner, D.: Recent progress on the Daugavet property. Ir. Math. Soc. Bull. 46, 77–97 (2001)

    Article  MathSciNet  Google Scholar 

  32. Werner, D.: The Daugavet equation for operators not fixing a copy of \(C[0,1]\). J. Oper. Theory 39, 89–98 (1998)

    MathSciNet  MATH  Google Scholar 

  33. Wojtaszczyk, P.: Some remarks on the Daugavet equation. Proc. Am. Math. Soc. 115, 1047–1052 (1992)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics Education, Dongguk University - Seoul, Seoul, 04620, Republic of Korea

    Han Ju Lee

  2. Research Institute for Natural Sciences, Hanyang University, Seoul, 04763, Republic of Korea

    Hyung-Joon Tag

Authors
  1. Han Ju Lee
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hyung-Joon Tag
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Han Ju Lee.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

H. J. Lee was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (NRF-2020R1A2C1A01010377). H.-J. Tag was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (NRF-2020R1A2C1A01010377, NRF-2020R1C1C1A01010133)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lee, H.J., Tag, HJ. Diameter two properties in some vector-valued function spaces. RACSAM 116, 17 (2022). https://doi.org/10.1007/s13398-021-01165-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s13398-021-01165-6

Keywords

Mathematics Subject Classification

Search

Navigation