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

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.