Contractibility of ultrapower of Frechet algebras

Abstract

The aim of this article is to study a number of relationship between Frechet algebra $\mathcal{A}$ and its ultrapower $(\mathcal{A}{)}_{\mathcal{U}}$. We give a characterization in some aspects such as locally bounded approximate identity. We consider the notion of contractibility of ultrapower of Frechet algebra, and we show that if $(\mathcal{A}{)}_{\mathcal{U}}$ has approximation property with good ultrafilter $\mathcal{U}$ then $\mathcal{A}$ is contractible if and only if $(\mathcal{A}{)}_{\mathcal{U}}$ is contractible.