Functors induced by Cauchy extension of C*-algebras

TL;DR

This paper introduces three functors on C*-algebras related to Cauchy extensions, analyzing their properties and showing that two are exact while one is normal exact, advancing the understanding of algebraic extensions.

Contribution

It defines and studies three new functors on C*-algebras, revealing their exactness properties and their roles in Cauchy extensions of algebras.

Findings

The functors $[ullet]_K$ and $rak{F}$ are exact.

The functor $rak{P}$ is normal exact.

Properties of these functors in the context of Cauchy extensions are established.

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.