Vol. 65, No. 1, 1976

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Well-behaved and totally bounded approximate identities for C0(X)

Robert Francis Wheeler

Vol. 65 (1976), No. 1, 261–269

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)λΛ such that (1) fλ1λ; and (2) if h C0(X), then limλhfλ 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.

Mathematical Subject Classification 2000
Primary: 46E25
Secondary: 54C40
Received: 30 September 1974
Revised: 14 February 1975
Published: 1 July 1976
Robert Francis Wheeler