U.P.B. Sci. Bull., Series A, Vol. 75, Iss. 3, 2013 ISSN 1223-7027 SECOND ORDER (σ, τ )-COHOMOLOGY OF TRIANGULAR BANACH ALGEBRAS Ali Jabbari 1 , Hasan Hosseinzadeh 2 Let A be a Banach algebra. In this paper, we introduce (σ, τ )-2-cocycle and (σ, τ )-coboundary maps on A, where σ and τ are homomorphisms on A. By applying these definitions, we introduce the second (σ, τ )-cohomology of triangular Banach algebras. Keywords: Banach module, Cohomology, Hochschild cohomology, Triangular Banach algebras. 2000 Mathematics Subject Classification: Primary 46H25, Secondary 16E40. 1. Introduction Let A be a Banach algebra, and let X be a Banach A-bimodule. A derivation is a linear map D : A −→ X such that D(ab)= a.D(b)+ D(a).b (a, b ∈ A). For x ∈ X , set ad x : a → a.x − x.a, A −→ X . Then ad x is the inner derivation induced by x. The linear space of bounded derivations from A into X denoted by Z 1 (A,X ) and the linear subspace of inner derivations denoted by N 1 (A,X ). We consider the quotient space H 1 (A,X )= Z 1 (A,X )/N 1 (A,X ), called the first Hochschild coho- mology group of A with coefficients in X . Let A and B be unital Banach algebras with units e A and e B , respectively. Suppose that M is a unital Banach A, B-bimodule. We define triangular Banach algebra T = [ A M B ] , with the sum and product being giving by the usual 2 × 2 matrix operations and internal module actions. The norm on T is ∥ [ a m b ] ∥ = ∥a∥ A + ∥m∥ M + ∥b∥ B . 1 Young Researchers and Elite Club, Ardabil Branch, Islamic Azad University, Ardabil, Iran, E-mail: jabbari al@yahoo.com & ali.jabbari@iauardabil.com 2 Department of Mathematics, Ardabil Branch, Islamic Azad University, Ardabil, Iran, E-mail: hasan hz2003@yahoo.com, corresponding author 59