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

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Let A be a Banach algebra and $$\phi $$ be a character on A. In this paper we consider the class $${\mathscr {S}}{\mathscr {M}}^{A}_{\phi }$$ of Banach A-bimodules X for which the module actions of A on X is given by $$a \cdot x = x \cdot a = \phi (a)x $$ ( $$a \in A, x \in X$$ ) and we study the first and second continuous Hochschild cohomology groups of A with coefficients in $$X\in {\mathscr {S}}{\mathscr {M}}^{A}_{\phi }$$ . We obtain some sufficient conditions under which $$H^1(A,X)=\lbrace 0 \rbrace $$ and $$H^2(A,X)$$ is Hausdorff, where $$X\in {\mathscr {S}}{\mathscr {M}}^{A}_{\phi }$$ . We also consider the property that $$H^1(A,X)=\lbrace 0 \rbrace $$ for every $$X\in {\mathscr {S}}{\mathscr {M}}^{A}_{\phi }$$ and get some conclusions about this property. Finally, we apply our results to some Banach algebras related to locally compact groups.