  Let $Y$ be a compact K\"ahler manifold. We show that the weak regularization
$K_n$ of Dinh and Sibony for the diagonal $\Delta_Y$ (see Section 2 for more
detail) is compatible with wedge product in the following sense:
  If $T$ is a positive $dd^c$-closed $(p,p)$ current and $\theta$ is a smooth
$(q,q)$ form then there is a sequence of positive $dd^c$-closed $(p+q,p+q)$
currents $S_n$ whose masses converge to 0 so that $-S_n\leq K_n(T\wedge
\theta)-K_n(T)\wedge \theta \leq S_n$ for all $n$.
  We also prove a result concerning the quasi-potentials of positive closed
currents. We give two applications of these results. First, we prove a
corresponding compatibility with wedge product for the pullback operator
defined in our previous paper. Second, we define an intersection product for
positive $dd^c$-closed currents. This intersection is symmetric and has a local
nature.
