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.
Categories: math.FA