The stereotype approximation property for the algebras \(\mathcal{C}(M)\) of continuous functions on metric spaces (Q2210390)

In the paper [\textit{S. S. Akbarov}, J. Math. Sci., New York 113, No. 2, 179--349 (2003; Zbl 1042.46002)], a Hausdorff locally convex space \(X\) over \(\mathbb C\) is called stereotype if the canonical map \(X\to X^{**}\) is a topological isomorphism, where the dual \(X^*\) is endowed with the topology of uniform convergence on precompact subsets of \(X\) (which is the same as being polar reflexive in the sense of [\textit{G. Köthe}, Topological vector spaces. I. Berlin: Springer (1969; Zbl 0179.17001)]. Let \((M,d)\) be a complete metric space and let \(\mathcal{C}_{\sharp}(M)\) be the algebra of continuous functions \(f:M\to \mathbb C\) with the usual pointwise multiplication and endowed with the topology of uniform convergence on compact subsets of \(M\). By virtue of a result from the author's mentioned paper, the pseudosaturation \(\mathcal{C}_{\sharp}(M)^{\Delta}\) of the space \(\mathcal{C}_{\sharp}(M)\) is a stereotype algebra -- it is here denoted as \(\mathcal{C}(M)\). The author proves that, for any space \(M\), the algebra \(\mathcal{C}(M)\) has the stereotype approximation property (in the sense of the above mentioned paper).