The First and Second Hochschild Cohomology Groups of Banach Algebras with Coefficients in Special Symmetric Bimodules

摘要

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