In this paper we give three functors $\mathfrak{ P}$, $[\cdot]_K$ and $\mathfrak{ F}$ on the category of C$^\ast$-algebras. The functor $\mathfrak{ P}$ assigns to each C$^\ast$-algebra $\mathcal{ A}$ a pre-C$^\ast$-algebra $\mathfrak{ P}(\mathcal{ A})$ with completion $[\mathcal{ A}]_K$. The functor $[\cdot]_K$ assigns to each C$^\ast$-algebra $\mathcal{ A}$ the Cauchy extension $[\mathcal{ A}]_K$ of $\mathcal{ A}$ by a non-unital C$^\ast$-algebra $\mathfrak{ F}(\mathcal{ A})$. Some properties of these functors are also given. In particular, we show that the functors $[\cdot]_K$ and $\mathfrak{ F}$ are exact and the functor $\mathfrak{ P}$ is normal exact.
Categories: math.OA math.FA