Document Type : Research Paper

Authors

Department of Mathematics, Faculty of Science, Arak University, 38156-8-8349, Arak, Iran.

Abstract

Let $(A,\| \cdot \|)$ be a real Banach algebra, a complex algebra $A_\mathbb{C}$ be a complexification of $A$ and $\| | \cdot \| |$ be an algebra norm on  $A_\mathbb{C}$  satisfying a simple condition together with the norm $\| \cdot \|$ on $A$.  In this paper we first show that $A^*$ is a real Banach $A^{**}$-module if and only if $(A_\mathbb{C})^*$ is a complex Banach $(A_\mathbb{C})^{**}$-module. Next  we prove that $A^{**}$ is $(-1)$-weakly  amenable if and only if $(A_\mathbb{C})^{**}$ is $(-1)$-weakly  amenable. Finally, we give some examples of real Banach algebras which their second duals of some them are and of others are not $(-1)$-weakly  amenable.

Keywords

Main Subjects

###### ##### References
[1] D. Alimohammadi and A. Ebadian, Hedberg's theorem in real Lipschitz algebras, Indian J. Pure Appl. Math., 32 (2010), pp. 1470-1493.

[2] D. Alimohammadi and T.G. Honary, Contractibility, amenability and weak amenability of real Banach algebras, J. Aanalysis, (9)(2001), pp. 69-88.

[3] R. Arens, The adjoint of a bilinear operation, Proc. Math. Amer. Soc., 2 (1951), pp. 839-848.

[4] W.G. Bade, P.C. Curtis, and H.G. Dales, Amenability and weak amenability for Beurling and Lipschitz algebras, Proc. London Math. Soc., 55 (1987), pp. 359-377.

[5] F.F. Bonsall and J. Duncan, Complete Normed Algebras, Springer Verlag, New York, 1973.

[6] H.G. Dales, Banach Algebras and Automatic Continuity, Oxford University Press, 2000.

[7] J. Duncan and S.A.R. Hosseinioun, The second dual of a Banach algebra, Proc. Roy. Soc. Edinburg Sect. A., 84 (1979), pp. 309-325.

[8] M. Eshaghi Gordji, S.A.R. Hosseinioun, and A. Valadkhani, On (-1)-weak amenability of Banach algebras, Math. Reports, 15 (65), (2013), pp. 271-279.

[9] T.G. Honary and S. Moradi, On the maximal ideal space of extended analytic Lipschitz algebras, Quaestiones Mathematicae, 30 (2007), pp. 349-353.

[10] S.A.R. Hosseinioun and A. Valadkhani, (-1)-Weak amenability of unitized Banach algebras, Europ. J. Pure Appl. Math., 9 (2016), pp. 231-239.

[11] S.A.R. Hosseinioun and A. Valadkhani, Weak and (-1)-weak amenability of second dual of Banach algebras, Int. J. Nonlinear Anal. Appl., 7 (2016), pp. 39-48.

[12] B.E. Johnson, Cohomology in Banach algebras, Mem. Amer. Math. Soc., 127 (1972).

[13] B.E. Johnson, Derivations from $L^1(G)$ into $L^1(G)$ and $L^infty (G)$, Proc. International conference on Harmonic Analysis, Luxembourg, (Lecture note in Math. Springer-Verlag), 1359 (1987), pp. 191-198.

[14] S.H. Kulkarni and B.V. Limaye, Gleason parts of real function algebras, Canad. J. Math., 33 (1981), pp. 181-200.

[15] S.H. Kulkarni and B.V. Limaye, Real Function Algebras, Marcel Dekker, Inc. New York, 1992.

[16] M. Mayghani and D. Alimohammadi, The Structure of ideals, point derivations, amenability and weak amenability of extended Lipschitz algebras, Int. J. Nonlinear Anal. Appl., 2017, pp. 389-404.

[17] A. Medghalchi and T. Yazdanpanah, Problems concerning n-weak amenability of a Banach algebra, Czechoslovak Math. J., 55 (2005), pp. 863-876.

[18] T.W. Palmer, Banach Algebras, the General Theory of *-Algebras, Vol. 1: Algebras and Banach Algebras, Cambridge University Press, Cambridge, 1994.

[19] D.R. Sherbert, Banach algebras of Lipschitz functions, Pacific J. Math., (13) (1963), pp. 1387-1399.

[20] D.R. Sherbert, The structure of ideals and point derivations in Banach algebras of Lipschitz functions, Trans. Amer. Math. Soc, 111 (1964), pp. 240-272.