  We prove that for a Banach algebra $A$ having a bounded
$\mathcal{Z}(A)$-approximate identity and for every $\bf[IN]$ group $G$ with
weight $w$ which is either constant on conjugacy classes or $w \geq 1$,
$\mathcal{Z}\big(L^1_w(G) \otimes^\gamma A\big) \cong \mathcal{Z}(L^1_w(G))
\otimes^\gamma \mathcal{Z}(A)$. As an application, we discuss the conditions
under which $\mathcal{Z}\big(L^1_w(G,A)\big)$ enjoys certain Banach algebraic
properties, for example, weak amenability, semisimplicity etc.
