Close-to-Regularity of Bounded Tri-Linear Maps

Document Type : Original Research Article

Authors

1 Independent Researcher.

2 Department of Mathematics and Applications, University of Mohaghegh Ardabili, Ardabi, Iran.

3 Department of Mathematics , University of Urmia, Urmia, Iran.

Abstract

Let f : X × Y × Z → W be a bounded tri-linear map on normed spaces. We say that f is close-to-regular when ft∗∗∗∗s = fs∗∗∗∗t and we say that f is Aron-Berner regular when all natural extensions are equal. In this manuscript, we give a simple criterion for the close-to-regularity of tri-linear maps. 

Keywords

References

  1. R . Arens, The adjoint of a bilinear operation, Proc. Amer. Math. Soc.2 , 839-848 (1951).
  2. N. Arikan, A simple condition ensuring the Arens regularity of bilinear mappings, Proc. Amer. Math. Soc. 84 , 525-532 (1982).
  3. R. M. Aron, P. Berner, A Hahn-Banach extension theorem for analytic mappings, Bull. Soc. Math. France. 106 3–24 (1978).
  4. R. M. Aron, P. Galindo, Weakly compact multilinear mappings, Proc. Edinb. Math. Soc. 40 181–192 (1997).
  5. R. M. Aron, P. Galindo, D. Garc´ıa and M. Maestre, Regularity and algebras of analytic functions in infinite dimensions, Trans. Amer. Math. Soc. 348 (2) 543–559 (1996).
  6. P. Civin and B. Yood, The second conjugate space of a Banach algebra as an algebra, Pacific. J. Math. 11, no. 3, 847-870 (1961).
  7. M. Eshaghi Gordji and M. Filali, Arens regularity of module actions, Studia Math 181, no. 3, 237-254 (2007).
  8. G. B. Folland, A course in abstract harmonic analysis, Crc Press, (1995).
  9. K. Haghnejad Azar, Arens regularity of bilinear forms and unital Banach module space, Bull. Iranian Math. Soc. 40, no. 2, 505-520 (Ref2014).
  10. A. A. Khosravi, H. R. E. Vishki and A.M. Peralta, Aron–Berner extensions of triple maps with application to the bidual of Jordan Banach triple systems, Linear Algebra Appl. 580, 436-463 (2019).
  11. S. Mohamadzadeh and H. R. E. Vishki, Arens regularity of module actions and the second adjoint of a drivation, Bull Austral. Mat. Soc, 77, no. 3, 465-476 (2008).
  12. A. Ulger, Weakly compact bilinear forms and Arens regularity, Proc. Amer, Math. Soc 101, no. 4, 697-704 (1987).
  13. A. Sheikhali, A. Ebadian and K. Haghnejad Azar, Regularity of bounded tri-linear maps and the fourth adjoint of a tri-derivation, Global Analysis and Discrete Mathematics, 5, no. 1, 51-65 (2020).
  14. N. J. Young, The irregularity of multiplication in group algebras, Quart. J. Math. Oxford 24, no. 1, 59-62 (1973).
Analytical and Numerical Solutions for Nonlinear Equations
Volume 6, Issue 1
August 2021
Pages 33-39

Files

History

  • Receive Date: 10 November 2020
  • Revise Date: 10 January 2021
  • Accept Date: 09 February 2021
  • Publish Date: 01 August 2021

Share

How to cite

Statistics