  We show that the following operation, $(f \star g)(x,y)=
\int_{-\infty}^{+\infty} f (x,t) g(t,y) dt$, is well defined on the weak
closure of the space of functions that are smooth over a compact of
$\mathbb{R}^2$. We establish that a subset of this weak closure has the
structure of a Fr\'echet space $\mathcal{D}$ on distributions. Invertible
elements of $\mathcal{D}$ form a dense subset of it and a Fr{\'e}chet-Lie group
for the operation $\star$. This product generalizes the convolution, Volterra
compositions of first and second type and Schwartz's bracket.
