Characterization of Some Biderivations on Triangular Banach Algebras

Abstract

Let $A$ and $B$ be unital Banach algebras‎, ‎$X$ be an unital $A-B-$module and $T$ be the triangular Banach algebra associated to $A‎, ‎B$ and $X$‎. The structure of some derivations on triangular Banach algebras was studied by some authors. ‏‎Note that despite the apparent similarity between derivations and biderivations and also inner derivations and inner biderivations‎, ‎there are fundamental differences between them‎. Although there are some studying of biderivations on triangular Banach algebras, any of them do not completely determine the structure of biderivations on triangular Banach algebras. In this paper, we ‎completely characterize biderivations and inner biderivations from $T\times T$ to $T^*$‎ and we show that the first bicohomology group $BH^1(T, T^*)$ is equal to $BH^1(A, A^*)\oplus BH^1(B, B^*)$‎‏.