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Normal structure and fixed point properties for some Banach algebras

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Abstract

In this paper, we investigate key assumptions on some Banach algebras to have quasi-weak normal structure (resp. \(\hbox {quasi-weak}^{\star }\) normal structure) and to prove in particular that fixed point property for Kannan mappings is satisfied on weakly compact convex subsets in this setting. We establish some conditions on a locally compact group G for which the Fourier and Fourier–Stieltjes–Banach algebras A(G) and B(G) have quasi-normal structure. In addition, we give some examples of Banach spaces X such that \({{\mathcal {L}}} (X)\) (the Banach algebra of bounded linear operators on X) and some of its closed two-sided ideals associated with Fredholm perturbations have fixed point properties. Finally, we give some comments and interesting questions related to this area.

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References

  1. Alspach, D.: A fixed point free nonexpansive map. Proc. Am. Math. Soc. 82, 423–424 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  2. Androulakis, G., Schlumprecht, T.: Strictly singular, non-compact operator exist on the space of Gowers and Maurey. J. Lond. Math. Soc. 64(3), 655–674 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Arazy, J.: More on convergence in unitary matrix spaces. Proc. Am. Math. Soc. 83, 44–48 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  4. Argyros, S.A., Haydon, R.: A hereditarily indecomposable \({\cal{L}}_\infty \) space that solves the scalar plus-compact problem. Acta Math. 206(1), 1–54 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Belluce, L.P., Kirk, W.A., Steiner, E.F.: Normal structure in Banach spaces. Pac. J. Math. 26, 433–440 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bourgain, J.: On the Dunford-Pettis property. Proc. Am. Math. Soc. 81(2), 265–272 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  7. Brodskii, M.S., Milman, D.P.: On the center of a convex set. Dokl. Akad. Nauk (SSSR) 59, 837–840 (1948)

    MathSciNet  Google Scholar 

  8. Bynum, W.L.: Normal structure of Banach spaces. Manuscr. Math. 11(3), 203–209 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  9. Calkin, J.W.: Two-sided ideals and congruences in the ring of bounded operators in Hilbert space. Ann. Math. 2(42), 839–873 (1941)

    Article  MathSciNet  MATH  Google Scholar 

  10. Carothers, N.L.: A short course on Banach space theory. In: London Math Soc Student Texts, vol. 64. Cambridge University Press, Cambridge (2005)

  11. Cho, K., Lee, C.: Banach-Saks property on the dual of Schlumprecht space. Kangweon-Kyungki Math. J. 6(2), 341–348 (1998)

    Google Scholar 

  12. Chu, C.H.: A note on scattered \(C^{\star }\)-algebras and the Radon-Nikodym property. J. Lond. Math. Soc. 2(3), 533–536 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  13. Diestel, J.: Geometry of Banach spaces-Selected Topics. Lecture Notes in Mathematics, vol. 465. Springer, Berlin (1975)

  14. Dowling, P.N., Lennard, C.J.: Every nonreflexive subspace of \(L_1([0, 1])\) fails the fixed point property. Proc. Am. Math. Soc. 125(2), 443–446 (1997)

    Article  MATH  Google Scholar 

  15. Dowling, P.N., Randrianantoanina, N.: Spaces of compact operators on a Hilbert space with the fixed point property. J. Func. Anal. 169, 111–120 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  16. van Dulst, D., Sims, B.: Fixed points of nonexpansive mappings and Chebyshev centers in Banach spaces with norms of type (KK). In: Banach Space Theory and Applications (Bucharest 1981), Lectures Notes in Mathematics, vol. 991, pp. 35–43. Springer, Berlin (1983)

  17. Dye, J., Khamsi, M.A., Reich, S.: Random products of contractions in Banach spaces. Trans. Am. Math. Soc. 325(1), 87–99 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  18. Enflo, P.: A counter example to the approximation problem in Banach spaces. Acta Math. 130, 309–317 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  19. Eymard, P.: L’algèbre de Fourier d’un groupe localement compact. Bull. Soc. Math. Soc. 92, 181–236 (1964)

    MathSciNet  MATH  Google Scholar 

  20. Fabian, M., Habala, P., Hájek, P., Santalucia, V.M., Pelant, J., Zizler, V.: Functional Analysis and Infinite Dimensional Geometry. Springer, New York (2001)

    Book  MATH  Google Scholar 

  21. Fendler, G., Leinert, M.: Separable \(C^{\star }\)-algebras and \(\text{ weak }^{\star }\) fixed point property. Probab. Math. Stat. 33(2), 233–241 (2013)

    MathSciNet  MATH  Google Scholar 

  22. Ferenczi, V.: A uniformly convex hereditarily indecomposable Banach space. Israel J. Math. 102(1), 199–225 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  23. Goebel, K., Kirk, W.A.: Topics in Metric Fixed Point Theory, vol. 28. Cambridge University Press, Cambridge (1990)

    Book  MATH  Google Scholar 

  24. Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Marcel Dekker, New York (1984)

    MATH  Google Scholar 

  25. Gohberg, I.C., Markus, A.S., Feldman, I.A.: Normally solvable operators and ideals associated with them (in Russian), Bul. Akad. \({\widehat{S}}\)tiince RSS Moldoven. 76(10), 51–70 (1960) (translation: Amer. Math. Soc. Transl. 2(61), 63-84 (1967)

  26. Gonzalez, M.: The perturbation classes problem in Fredholm theory. J. Func. Anal. 200(1), 65–70 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  27. Gowers, T.W., Maurey, B.: The unconditional basic sequence problem. J. Am. Math. Soc. 6, 851–874 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  28. Gowers, T.W.: A new dichotomy for Banach spaces. Geom. Func. Anal. 6(6), 1083–1093 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  29. Grothendieck, A.: Sur les applications linéaires faiblement compactes d’espaces de type \(C(K)\). Can. J. Math. 5, 129–173 (1953)

    Article  MATH  Google Scholar 

  30. Holub, J.R.: On subspaces of separable norm ideals. Bull. Am. Math. Soc. 79(2), 446–448 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  31. Huff, R.: Banach spaces which are nearly uniformly convex. Rocky Mt. J. Math. 10(4), 743–749 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  32. Kaniuth, E., Lau, A.T.: Fourier and Fourier–Stieljes Algebras on Locally Compact Groups. Mathematical Surveys and Monographs, vol. 231. American Mathematical Society (2018)

  33. Karlovitz, L.A.: Existence of fixed points of nonexpansive mappings in a space without normal structure. Pac. J. Math. 66(1), 153–159 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  34. Khamsi, M.A., Kirk, W.A.: An introduction to metric spaces and fixed point theory. In: Pure and Applied Mathematics: A Wiley-Interscience Series of Texts, Monographs and Tracts (2001)

  35. Kirk, W.A.: On successive approximations for nonexpansive mappings. Glasg. Math. J. 12(1), 6–9 (1971)

    Article  MATH  Google Scholar 

  36. Kirk, W.A.: A fixed point theorem of mappings which do not increase distance. Am. Math. Mon. 72(9), 1004–1006 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  37. Kutzarova, D., Lin, P.K.: Remarks about Schlumprecht space. Proc. Am. Math. Soc. 128(7), 2059–2068 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  38. Landes, T.: Permanence properties of normal structure. Pac. J. Math. 110(1), 125–143 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  39. Lau, A.T., Mah, P.F., Ülger, A.: Fixed point property and normal structure for Banach spaces associated to locally compact groups. Proc. Am. Math. Soc. 125(7), 2021–2027 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  40. Lau, A.T., Mah, P.F.: Normal structure in dual Banach spaces associated with a locally compact group. Trans. Am. Math. Soc. 310(1), 341–353 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  41. Lau, A.T., Ülger, A.: Some geometric properties on the Fourier and Fourier Stieljes algebras of locally compact groups, Arens regularity and related problems. Trans. Am. Math. Soc. 337(1), 321–359 (1993)

    MATH  Google Scholar 

  42. Lau, A.T., Mah, P.F.: Quasi-normal structures for certain spaces of operators on a Hilbert space. Pac. J. Math. 121(1), 109–118 (1986)

    Article  MATH  Google Scholar 

  43. Lau, A.T.: Fixed Point and Related Properties for Fourier and Fourier–Stieljes Algebra of a Locally Compact Groups, Nonlinear Analysis and Applications: to V. Lakshmikantham on his 80th Birthday, vol. 1, 2, pp. 755–765. Kluwer Academic Publishers, Dordrecht (2003)

  44. Lau, A.T.: Normal structure and common fixed point properties for semigroups of nonexpansive mappings in Banach spaces. Fixed Point Theory Appl. 2010, 14 (2010)

    MathSciNet  MATH  Google Scholar 

  45. Lim, T.C.: Asymptotic centers and nonexpansive mappings in conjugate Banach spaces. Pac. J. Math. 90(1), 135–143 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  46. Lin, P.K.: Unconditional bases and fixed points of nonexpansive mappings. Pac. J. Math. 116(1), 69–76 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  47. Maurey, B.: Points fixes des contractions sur un convexe fermé de \(L^{1}\), Séminaire d’analyse fonctionnelle, École polytechnique, Palaiseau (1980–1981)

  48. Reich, S.: Kannan’s fixed point theorem. Boll. Un. Math. Ital. 4, 1–11 (1971)

    MathSciNet  MATH  Google Scholar 

  49. Reich, S.: Fixed points of contractive functions. Boll. Un. Math. Ital. 5, 26–42 (1972)

    MathSciNet  MATH  Google Scholar 

  50. Reich, S.: The fixed point property for nonexpansive mappings I. Am. Math. Mon. 83, 266–268 (1976)

    Article  MATH  Google Scholar 

  51. Reich, S.: The fixed point property for nonexpansive mappings II. Am. Math. Mon. 87, 292–294 (1980)

    Article  MATH  Google Scholar 

  52. Roux, D., Zanco, C.: Kannan maps in normed spaces. Rend. Sci. Fis. Math. Nat. 65, 252–258 (1978)

    MathSciNet  MATH  Google Scholar 

  53. Schlumprecht, T.: An arbitrary distortable Banach space. Israel J. Math. 76(1), 81–95 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  54. Smyth, M.A.: On the failure of close-to-normal structure type conditions and pathological Kannan mappings. Proc. Am. Math. Soc. 124(10), 3063–3069 (1996)

    Article  MATH  Google Scholar 

  55. Soardi, P.M.: Struttura quasi normale e teoremi di punto unito. Rend. 1st Math. Univ Trieste, vol. 4, pp. 105–114 (1972)

  56. Weis, L.: On perturbations of Fredholm operators in \(L_p(\mu )\)-spaces. Proc. Am. Math. Soc. 67(2), 287–292 (1977)

    Google Scholar 

  57. Wong, C.S.: Close-to-normal structure and its applications. J. Func. Anal. 16(4), 353–358 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  58. Wong, C.S.: On Kannan maps. Proc. Am. Math. Soc. 47(1), 105–111 (1975)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

I would like to thank the distinguished professor Anthony To-Ming Lau for his invitation at the University of Alberta during the period May 11–May 24, 2019, I express my great respect and admiration to him and to this institution. I am indebted to Canadian government for its noble initiative to facilitate the mobility of foreign researchers and encouraging them to share knowledge with their counterparts in Canada. Also, I am grateful to the anonymous referees for their valuable remarks and suggestions to improve the manuscript.

Funding

This work is supported by the research team RPC (Controllability and Perturbation Results) in the laboratory of Informatics and Mathematics (LIM) at the university of Souk-Ahras (Algeria).

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  1. Laboratory of Informatics and Mathematics, University of Souk-Ahras, P.O.Box 1553, 41000, Souk-Ahras, Algeria

    Abdelkader Dehici

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Dehici, A. Normal structure and fixed point properties for some Banach algebras. J. Fixed Point Theory Appl. 22, 7 (2020). https://doi.org/10.1007/s11784-019-0743-6

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