Addendum to “Amenability and weak amenability of second conjugate Banach algebras”

Ghahramani, F.; Loy, R.; Willis, G.

doi:10.1090/proc/15009

Addendum to “Amenability and weak amenability of second conjugate Banach algebras”
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by F. Ghahramani, R. J. Loy and G. A. Willis PDF

Proc. Amer. Math. Soc. 148 (2020), 4573-4575 Request permission

Abstract:

The purpose of this note is to show that if $A$ is a Banach algebra with the continuous dual space $A^{*}$ and $D : A \rightarrow A^{*}$ is a weakly compact derivation, then $D^{**} : A^{**} \rightarrow A^{***}$ is also a derivation, where $A^{**}$ has the first (or second) Arens product and $A^{***}$ is viewed as the dual module of the Banach algebra $A^{**}.$

References

Massoud Amini, Morteza Essmaili, and Mahmoud Filali, The second transpose of a derivation and weak amenability of the second dual Banach algebras, New York J. Math. 22 (2016), 265–275. MR 3484685
H. G. Dales, A. Rodríguez-Palacios, and M. V. Velasco, The second transpose of a derivation, J. London Math. Soc. (2) 64 (2001), no. 3, 707–721. MR 1865558, DOI 10.1112/S0024610701002496
Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
F. Ghahramani, R. J. Loy, and G. A. Willis, Amenability and weak amenability of second conjugate Banach algebras, Proc. Amer. Math. Soc. 124 (1996), no. 5, 1489–1497. MR 1307520, DOI 10.1090/S0002-9939-96-03177-2