Abstract :
Let A1, A2 be unital Banach algebras and X be an A1 ?
A2? module. Applying the concept of module maps, (inner) module
generalized derivations and generalized first cohomology groups, we
present several results concerning the relations between module generalized
derivations from Ai into the dual space A
?
i (for i = 1, 2) and
such derivations from the triangular Banach algebra of the form T :=
A1 X
0 A2
into the associated triangular T ? bimodule T ?
of the
form T ?
:=
A
?
1 X
?
0 A
?
2
. In particular, we show that the so-called
generalized first cohomology group from T to T ?
is isomorphic to the
directed sum of the generalized first cohomology group from A1 to A
?
1
and the generalized first cohomology group from A2 to A
?
2. Finally,
Inspiring the above concepts, we establish a one to one corresponding
between weak (resp. ideal) generalized amenability of T and those
amenability of Ai (i = 1, 2).
