Contractibility of ultrapower of Frechet algebras

TL;DR

This paper investigates the relationship between a Frechet algebra and its ultrapower, focusing on conditions under which contractibility is preserved, particularly involving approximation properties and ultrafilters.

Contribution

It provides a characterization of contractibility for ultrapowers of Frechet algebras and establishes conditions linking the properties of the algebra and its ultrapower.

Findings

If the ultrapower has the approximation property with a good ultrafilter, then the original algebra's contractibility is equivalent to that of its ultrapower.

The paper characterizes the relationship between the algebra and its ultrapower regarding locally bounded approximate identities.

It advances understanding of how ultrapower constructions affect algebraic properties like contractibility.

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.