Let X be a locally compact Hausdorff space. The space C_0(X) of complex-valued continuous functions on X that vanish at infinity (defined in the article on local compactness) form a commutative C*-algebra C_0(X) under pointwise multiplication and addition. The involution is pointwise conjugation. C_0(X) has a multiplicative unit element if and only if X is compact. As does any C*-algebra, C_0(X) has an approximate identity. In the case of C_0(X) this is immediate: consider the directed set of compact subsets of X, and for each compact K let f_K be a function of compact support which is identically 1 on K. Such functions exist by the Tietze extension theorem, which applies to locally compact Hausdorff spaces. Any such sequence of functions \{f_K\} is an approximate identity.
