On asymmetry of topological centers of the second duals of Banach algebras

Abstract:

Let $\mathfrak {A}$ be a Banach algebra with a bounded approximate identity and let $Z_{1}(\mathfrak {A}^{**})$ and $Z_{2}(\mathfrak {A}^{**})$ be the left and right topological centers of $\mathfrak {A}^{**}$. It is shown that i) $\mathfrak {A}^{*}\mathfrak {A} = \mathfrak {A} \mathfrak {A}^{*}$ is not sufficient for $Z_{1}(\mathfrak {A}^{**}) = Z_{2}(\mathfrak {A}^{**})$; ii) the inclusion $\hat {\mathfrak {A}} Z_{1}(\mathfrak {A}^{**}) \subseteq \hat {\mathfrak {A}}$ is not sufficient for $Z_{2}(\mathfrak {A}^{**}) \hat {\mathfrak {A}} \subseteq \hat {\mathfrak {A}}$; iii) $Z_{1}(\mathfrak {A}^{**}) = Z_{2}(\mathfrak {A}^{**}) = \hat {\mathfrak {A}}$ is not sufficient for $\mathfrak {A}$ to be weakly sequentially complete. These results answer three questions of Anthony To-Ming Lau and Ali Ülger.