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LIFTING DERIVATIONS AND n-WEAK AMENABILITY OF THE SECOND DUAL OF A BANACH ALGEBRA

  • S. BAROOTKOOB (a1) and H. R. EBRAHIMI VISHKI (a2)

Abstract

We show that for n≥2, n-weak amenability of the second dual 𝒜** of a Banach algebra 𝒜 implies that of 𝒜. We also provide a positive answer for the case n=1, which sharpens some older results. Our method of proof also provides a unified approach to give short proofs for some known results in the case where n=1.

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Copyright

Corresponding author

For correspondence; e-mail: vishki@um.ac.ir

Footnotes

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This research was supported by a grant from Ferdowsi University of Mashhad (No. MP88107VIS).

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References

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[1]Arens, A., ‘The adjoint of a bilinear operation’, Proc. Amer. Math. Soc. 2 (1951), 839848.
[2]Bade, W. G., Curtis, P. C. and Dales, H. G., ‘Amenability and weak amenability for Beurling and Lipschitz algebras’, Proc. London Math. Soc. 55 (1987), 359377.
[3]Dales, H. G., Ghahramani, F. and Grønbæk, N., ‘Derivations into iterated duals of Banach algebras’, Studia Math. 128(1) (1998), 1954.
[4]Dales, H. G., Rodrigues-Palacios, A. and Velasco, M. V., ‘The second transpose of a derivation’, J. London Math. Soc. 64(2) (2001), 707721.
[5]Eshaghi Gordji, M. and Filali, M., ‘Weak amenability of the second dual of a Banach algebra’, Studia Math. 182(3) (2007), 205213.
[6]Ghahramani, F. and Laali, J., ‘Amenability and topological centres of the second duals of Banach algebras’, Bull. Aust. Math. Soc. 65 (2002), 191197.
[7]Ghahramani, F., Loy, R. J. and Willis, G. A., ‘Amenability and weak amenability of the second conjugate Banach algebras’, Proc. Amer. Math. Soc. 124 (1996), 14891497.
[8]Jabbari, A., Moslehian, M. S. and Vishki, H. R. E., ‘Constructions preserving n-weak amenability of Banach algebras’, Math. Bohem. 134(4) (2009), 349357.
[9]Johnson, B. E., ‘Weak amenability of group algebras’, Bull. London Math. Soc. 23 (1991), 281284.
[10]Mohammadzadeh, S. and Vishki, H. R. E., ‘Arens regularity of module actions and the second adjoint of a derivation’, Bull. Aust. Math. Soc. 77 (2008), 465476.
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LIFTING DERIVATIONS AND n-WEAK AMENABILITY OF THE SECOND DUAL OF A BANACH ALGEBRA

  • S. BAROOTKOOB (a1) and H. R. EBRAHIMI VISHKI (a2)

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