On Weak Generalized Amenability of Triangular Banach Algebras

Let a1, a2 be unital banach algebras and x be an a1 ?a2? module. applying the concept of module maps, (inner) modulegeneralized derivations and generalized first cohomology groups, wepresent several results concerning the relations between module generalizedderivations from ai into the dual space a?i (for i = 1, 2) ands
uch derivations from the triangular banach algebra of the form t :=a1 x0 a2into the associated triangular t ? bimodule t ?of theform t ?:=
a?1 x?0 a?2. in particular, we show that the so-calledgeneralized first cohomology group from t to t ?is isomorphic to thedirected sum of the generalized first cohomology group from a1 to a?1and the generalized first cohomology group from a2 to a?2. finally,inspiring the above concepts, we establish a one to one correspondingbetween weak (resp. ideal) generalized amenability of t and thoseamenability of ai (i = 1, 2).