Let X denote a locally
compact Hausdorff space, C0(X) the Banach algebra of continuous complex-valued
functions on X which vanish at infinity. An approximate identity for C0(X) is a net
(fλ)λ∈Λ such that (1) ∥fλ∥≦ 1∀λ; and (2) if h ∈ C0(X), then limλ∥hfλ− h∥ = 0.
Here the norm is the sup norm, and multiplication is the usual pointwise
product.
This paper contains an analysis of approximate identities for C0(X) of two special
types: totally bounded in the strict topology, and well-behaved in the sense
of Taylor. In each case, existence of an approximate identity of the stated
type is shown to be equivalent to paracompactness of X. A constructive,
somewhat lengthy proof of the first equivalence has been given by Collins and
Fontenot; here a short nonconstructive proof is presented. That well-behaved
implies paracompact is shown using a set-theoretic lemma of Hajnal. In
the course of the argument certain spaces X which can be embedded in
Stone-Čech compactifications of discrete spaces are considered. Using a result of
Rosenthal on relatively disjoint families of measures, it is shown that the strict
topology on C∗(X) is the Mackey topology for some of these X, not all
of which are paracompact. This indicates that σ-compact spaces can be
pasted together in fairly complicated ways while still retaining the Mackey
property.
