A construction of a bounded approximate A construction of a bounded approximate identity for a commutative C*-algebra A.

******************************

Let A = C0(X) be a commutative C*-algebra. Consider the set L consisting of all compact subsets of X. (L , Í ) is a directed set. For each compact subset K of X, by Urysohn's lemma, there exists a function fK Î C0(X) equal to 1 on K satisfying 0 £ f £ 1. For each g Î C0(X) and given e > 0, K0 = {x Î X; |g(x)| ³ e} is compact. Hence for all K Ê K0, ||fKg-g||¥ = supx Î X|fK(x)g(x)-g(x)| < e. Therefore limK Î LfKg = g, Thus (fK)K Î L is a bounded aproximate identity for A.


File translated from TEX by TTH, version 2.70.
On 22 Feb 2001, 00:18.