A construction of a bounded approximate identity for a commutative C^{∗}-algebra A.

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Let A = C_{0}(X) be a commutative C^{∗}-algebra. Consider the set Λ consisting of all compact subsets of X. (Λ, ⊆) is a directed set. For each compact subset K of X, by Urysohn’s lemma, there exists a function f_{K} ∈ C_{0}(X) equal to 1 on K satisfying 0 ≤ f ≤ 1. For each g ∈ C_{0}(X) and given ε > 0, K_{0} = {x ∈ X; |g(x)| ≥ ε} is compact. Hence for all K ⊇ K_{0}, ∥f_{K}g − g∥_{∞} = sup_{x∈X} |f_{K}(x)g(x) − g(x)| < ε. Therefore lim_{K∈Λ} f_{K}g = g, Thus (f_{K})_{K∈Λ} is a bounded aproximate identity for A.