A pre-dual for the convolution algebra $\mathcal D^\prime(\Gamma+)$

Let $\mathcal D'(\Gamma+)$ denote the subspace of $\mathcal D(\mathbb R^n)'$ consisting of all distributions whose support is contained in a translate of the cone $\Gamma$. We construct a locally convex space $\mathcal F(\Gamma+)$ consisting of $C^\infty$-functions on $\mathbb R^n$ such that $\mathcal D'(\Gamma+)$ is the dual of $\mathcal F(\Gamma+)$. We then discuss certain natural topologies on $\mathcal F(\Gamma+)$ and on $\mathcal D'(\Gamma+)$.