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.

中文翻译：

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具有特殊对称双模系数的Banach代数的第一和第二个Hochschild同调群

假设A为Banach代数，\（\ phi \）为A上的字符。在本文中，我们考虑Banach A -bimodules X的类\（{\ mathscr {S}} {\ mathscr {M}} ^ {A} _ {\ phi} \），其中A对X的模块作用为由下式给出\（一个\ CDOT X = X \ CDOT一个= \披（a）中X \） （\（一个\在A，X \在X \） ），我们研究的第一和第二连续尔德上同调群甲的系数为\（X \ in {\ mathscr {S}} {\ mathscr {M}} ^ {A} _ {\ phi} \）。我们获得一些足够的条件，其中\（H ^ 1（A，X）= \ lbrace 0 \ rbrace \）和\（H ^ 2（A，X）\）是Hausdorff，其中\（X \ in {\ mathscr {S}} {\ mathscr {M}} ^ {A} _ {\ phi} \）。我们还考虑以下属性：对于{\ mathscr {S}} {\ mathscr {M}} ^ {A} _中的每个\（X \ ），每个\（X \ ）都具有\（H ^ 1（A，X）= \ lbrace 0 \ rbrace \）{\ phi} \），并获得有关此属性的一些结论。最后，我们将结果应用于与局部紧致群有关的一些Banach代数。