| Abstract: | Let \({T}\) be a homomorphism from a Banach algebra \({B}\) to a Banach algebra \({A}\). The Cartesian product space \({A\times B}\) with \({T}\)-Lau multiplication and \({\ell^1}\)-norm becomes a new Banach algebra \({A\times _T B}\). We investigate the notions such as approximate amenability, pseudo amenability, \({\phi}\)-pseudo amenability, \({\phi}\)-biflatness and \({\phi}\)-biprojectivity for Banach algebra \({A\times_T B}\). We also present an example to show that approximate amenability of \({A}\) and \({B}\) is not stable for \({A\times _TB}\). Finally we characterize the double centralizer algebra of \({A\times _T B}\) and present an application of this characterization. |
