Approximate identities and paracompactness

Fontenot, R. A.; Wheeler, R. F.

doi:10.1090/S0002-9939-1987-0870777-3

Approximate identities and paracompactness
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by R. A. Fontenot and R. F. Wheeler

Proc. Amer. Math. Soc. 99 (1987), 232-236

DOI: https://doi.org/10.1090/S0002-9939-1987-0870777-3

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Abstract:

Let $X$ denote a locally compact Hausdorff space and ${C_b}(X)$ the algebra of continuous complex-valued functions on $X$. The main result of this paper is that $X$ is paracompact if and only if ${C_0}(X)$, the subalgebra of ${C_b}(X)$ consisting of functions which vanish at infinity, has an approximate identity which is a relatively compact subset of ${C_b}(X)$ for the weak topology of the pairing of ${C_b}(X)$ with its strict topology dual.

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