Weak Centrality for Certain Tensor Products of $C^{\ast }$-algebras

In this article, we discuss the weak centrality of the tensor product $A{\otimes }_{\alpha }B$ of $C^{\ast }$-algebras $A$ and $B$ in terms of the weak centrality of $A$ and $B$, where $\alpha$ is either the Haagerup or the Banach space projective tensor product. In the due course, we also identify the largest weakly central ideal of $A{\otimes }_{\alpha }B$ in certain cases. Centralilty and quasi-centrality of these tensor products are also discussed.