Abstract
Let A be a Banach algebra, A*, A^{**}
and A^{***} be its first,
second and third dual, respectively. Let
R:A^{***}⟶A* be the
restriction map, J:
A*⟶A^{***} be the canonical
injection and Λ: A^{***}→A^{***} be
the composition of R and J. Let D:A⟶
A* be a continuous derivation and
D^{''}:A^{**}⟶ A^{***} be its
second transpose. We obtain a necessary and sufficient condition for
Λ∘D^{''}:A^{**}⟶
(A^{**})* to be a derivation. We apply this to prove
some results on weak amenability of second dual Banach
algebras.

Author information
Massoud Amini:
Department of Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, 14115134 Tehran, Iran and School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O.Box 193955746, Tehran, Iran
mamini@modares.ac.ir
Morteza Essmaili:
Department of Mathematics, Faculty of Mathematical and Computer Science, Kharazmi University, 50 Taleghani Avenue, 15618 Tehran, Iran
m.essmaili@khu.ac.ir
Mahmoud Filali:
Department of Mathematical Sciences, University of Oulu, P.O. Box 3000, Oulu 90014, Finland
mahmoud.filali@oulu.fi
