Let A be a Banach algebra and φ be a character on A. In this paper we consider the class SMAφ of Banach A-bimodules X for which the module actions of A on X is given by a · x = x · a = φ(a)x (a ∈ A, x ∈ X) and we study the first and second continuous Hochschild cohomology groups of A with coefficients in X ∈ SMAφ . We obtain some sufficient conditions under which H1(A, X) = {0} and H2(A, X) is Hausdorff, where X ∈ SMAφ . We also consider the property that H1(A, X) = {0} for every X ∈SMAφ and get some conclusions about this property. Finally, we apply our results to some Banach algebras related to locally compact groups.
