  In this paper, we study operator algebraic properties of the reduced and von
Neumann algebraic versions of the free wreath products $\mathbb G \wr_* S_N^+$,
where $\mathbb G$ is a compact matrix quantum group. Based on recent result on
their corepresentation theory by Lemeux and Tarrago, we prove that $\mathbb G
\wr_* S_N^+$ is of Kac type whenever $\mathbb G$ is, and that the reduced
version of $\mathbb G \wr_* S_N^+$ is simple with unique trace state whenever
$N \geq 8$. Moreover, we prove that the reduced von Neumann algebra of $\mathbb
G \wr_* S_N^+$ does not have property $\Gamma$.
