Character Amenability of the Intersection of Lipschitz Algebras

Abtahi, Fatemeh ; Azizi, Mohsen ; Rejali, Ali

Canadian mathematical bulletin, Tome 60 (2017) no. 4, pp. 673-689

Résumé

Let $(X,d)$ be a metric space and let $J\subseteq [0,\infty )$ be nonempty. We study the structure of the arbitrary intersections of Lipschitz algebras and define a special Banach subalgebra of ${{\cap }_{\gamma \in J\,}}\text{Li}{{\text{p}}_{\gamma \,}}X$ , denoted by $\text{ILi}{{\text{p}}_{J}}X$ . Mainly, we investigate the $C$ -character amenability of $\text{ILi}{{\text{p}}_{J}}X$ , in particular Lipschitz algebras. We address a gap in the proof of a recent result in this field. Then we remove this gap and obtain a necessary and sufficient condition for $C$ -character amenability of $\text{ILi}{{\text{p}}_{J}}X$ , specially Lipschitz algebras, under an additional assumption.

Détail

DOI : 10.4153/CMB-2017-039-8

Mots-clés : 46H050, 46J10, 11J83, amenability, character amenability, Lipschitz algebra, metric space

Bibliographie

[1] [1] Abtahi, F., Amini, H. G., Lotfi, H. A., and Rejali, A., An arbitrary intersection ofLp-spaces. Bull. Aust. Math. Soc. 85(2012), no. 3, 433–445. Google Scholar | DOI

[2] [2] Abtahi, F., Amini, H. G., Lotfi, H. A., and Rejali, A., Some intersections of the weighted Lp-spaces. Abstr. Appl. Anal., Article ID 986857, 2013. Google Scholar

[3] [3] Abtahi, F., Amini, H. G., Lotfi, H. A., and Rejali, A., Some intersections ofLorentz spaces. Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 78(2016), no. 3, 73–82. Google Scholar

[4] [4] Bade, W. G., Curtis, P. C. Jr., and Dales, H. G., Amenability and weak amenability for Beurling and Lipschitz algebras. Proc. London Math. Soc. 55(1987), no. 2, 359–377. Google Scholar | DOI

[5] [5] Bishop, E. R., Generalized Lipschitz algebras. Canad. Math. Bull. 12(1969), 1–19. Google Scholar | DOI

[6] [6] Dashti, M., R. Nasr Isfahani, and S. Soltani Renani, Character amenability of Lipschitz algebras. Canad. Math. Bull., 57(2014), no. 1, 37–41. Google Scholar | DOI

[7] [7] Fukui, K. and T. Nakamura, A topological property of Lipschitz mappings. Topology Appl. 148(2005), 143–152. Google Scholar | DOI

[8] [8] E. Gourdeau, , Amenability ofBanach algebras. Math. Proc. Cambridge Philos. Soc. 105(1989), no. 2, 351–355. Google Scholar | DOI

[9] [9] E. Gourdeau, , Amenability ofLipschitz algebras. Math. Proc. Cambridge Philos. Soc. 112(1992), no. 3, 581–588. Google Scholar | DOI

[10] [10] Hu, Z., Monfared, M. S., and T. Traynor, On character amenable Banach algebras. Studia Math. 193(2009), no. 1, 53–78. Google Scholar | DOI

[11] [11] Jimenez-Vargas, A. and M. Villegas-Vallecillos, Homomorphisms on real-valued little Lipschitz function spaces. Topology Appl. 156(2009), 2908–2913. Google Scholar | DOI

[12] [12] Kaniuth, E., Lau, A. T., and J. Pym, On f-amenability ofBanach algebras. Math. Proc. Cambridge Philos. Soc. 144(2008), no. 1, 85–96. Google Scholar | DOI

[13] [13] Kaniuth, E., Lau, A. T., and J. Pym, On character amenability ofBanach algebras. J. Math. Anal. Appl. 344(2008), no. 2, 942–955. Google Scholar | DOI

[14] [14] Miyata, T. and T. Watanabe, Lipschitz functions and approximate resolutions. Topology Appl. 122(2002), 353–375. Google Scholar | DOI

[15] [15] Monfared, M. S., Character amenability ofBanach algebras. Math. Proc. Cambridge Philos. Soc. 144(2008), no. 3, 697–706. Google Scholar | DOI

[16] [16] Runde, V., Lectures on amenability. Lecture Notes in Mathematics, 1774, Springer-Verlag, Berlin, 2002. Google Scholar | DOI

[17] [17] Sherbert, D. R., Banach algebras of Lipschitz functions. Pacific J. Math. 13(1963), no. 4,1387-1399. Google Scholar | DOI

[18] [18] Sherbert, D. R., The structure of ideals andpoint derivations in Banach algebras of Lipschitz functions. Trans. Amer. Math. Soc. 111(1964), 240–272. Google Scholar | DOI

[19] [19] Zhang, Y., Weak amenability of a class ofBanach algebras. Canad. Math. Bull. 44(2001), no. 4, 504–508. Google Scholar | DOI