Stereotype approximation property for the algebras $C(M)$ of continuous functions on metric spaces

TL;DR

This paper proves that the algebra of continuous functions on any complete metric space possesses the stereotype approximation property, extending the understanding of approximation properties in the category of stereotype spaces.

Contribution

It establishes that $C(M)$ has the stereotype approximation property for any complete metric space, broadening the class of spaces known to have this property.

Findings

$C(M)$ has the stereotype approximation property for all complete metric spaces.

The result applies even when $M$ is not locally compact.

This advances the theory of approximation properties in stereotype spaces.

Abstract

In his previous works the author introduced the notion of the stereotype approximation property as an analog of the classical approximation property transferred to the category ${\tt Ste}$ of stereotype spaces. It is known, that the study of this property is much more difficult than in the classical theory, since the spaces of operators in the category ${\tt Ste}$ are defined in a more complicated way. In this note, it is proved that the algebra $C(M)$ of continuous functions on an arbitrary complete (not necessarily locally compact) metric space $M$ has the stereotype approximation property.