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

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.