$C(\alpha )$ preserving operators on $C(K)$ spaces

Abstract

Let A: C(K) -» X be a bounded linear operator where if is a compact Hausdorff space and X is a separable Banach space. Sufficient conditions are given for A to be an isomorphism (into) when restricted to a subspace Y of C(K), such that Y is isometrically isomorphic to a space C(a) of continuous functions on the space of ordinal numbers less than or equal to the countable ordinal a. 1. For a compact Hausdorff space K, C(K) will denote the Banach space of continuous real-valued functions on K with the supremum norm. The Greek letters a, s and y will be reserved for countable ordinal numbers. We consider the set [0, a] = {y:0 X > 0. For each countable ordinal a we will define a family Pa(X, B) of open subsets G and associated measures ju,G G B. Let P0(X, B) = {(G, jttG): G is an open set in K, ixG G B and | /?G | (G) > X}. If a s + 1, then Pa(X, B) = {(G, nc): (G, jiiG) G P0(X, B) and there is a disjoint sequence {G„} of open subsets of G and associated measures /xG such that (G„, ftG ) G Ps(X, B), ju,Gn -> juG and UnG C G). If a is a limit ordinal,then ^(X, B) = {(G, /xG) G P0(X, B): there is a disjoint sequence of open subsets {Gn} of G with associated measures /xG such that fic — nc, UnG„ C G and (G„, jUG ) G .Pa(X, B) where an/a). The notation a„ /a means that a„ is an increasing sequence of ordinals with a = sup„ an. A simple example illustrating the sets Pa(X, B) is contained in §4. Received by the editors March 20, 1979 and, in revised form, July 30, 1981. 1980 Mathematics Subject Classification. Primary 46E15. 'Research on this paper was partially supported by the National Science Foundation grant number MCS 77-01690. ©1982 American Mathematical Society O0O2-9939/82/OO0O-O726/SO5.OO