Functors induced by Cauchy extension of C*-algebras

Abstract

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.

arXiv comment24 pages