Arens regularity of module actions and weak amenability of Banach algebras

Arens Regularity of Projective Tensor Product

“…Similarly, B * * is a Banach A * * -bimodule with the module actions Suppose that A is a Banach algebra and B is a Banach A-bimodule. Since B * * is a Banach A * * -bimodule, where A * * is equipped with the first Arens product, we define the topological center of the right module action of A * * on B * * as follows: In this way, we write Z ℓ B * * (A * * ) = Z(π ℓ ), Z r A * * (B * * ) = Z(π t ℓ ) and Z r B * * (A * * ) = Z(π t r ), where π ℓ : A × B → B and π r : B × A → B are the left and right module actions of A on B, for more information related to the Arens regularity of module actions on Banach algebras, see [2,4,9,10]. If we set B = A, we write Z ℓ A * * (A * * ) = Z 1 (A * * ) = Z ℓ 1 (A * * ) and Z r A * * (A * * ) = Z 2 (A * * ) = Z r 2 (A * * ), for more information see [12].…”