The second dual of C(X)

Christine Ann Shannon

Vol. 72 (1977), No. 1, 237–253

DOI: 10.2140/pjm.1977.72.237

Abstract

In this paper, we undertake a study of the order dual, denoted M, of the radon measures of compact support on a locally compact space X. In the case that X is realcompact, M is the second (order) dual of the space of continuous functions on X, C(X). We define the sublattice of semi-continuous elements, S(X), and prove that each member of M is dominated by a member of S(X). It follows that the ideal generated by S(X) in M is all of M. On the other hand, the ideal generated by C(X) in M is all of M if and only if X is a cb-space.

Finally, we show that S(X) and C(X) can be identified in M as certain spaces of multiplication operators which are continuous with respect to certain weak topologies. This extends the work of J. Mack, who first characterized M as the (continuous) multiplication operators on the Radon measures.

Mathematical Subject Classification 2000

Primary: 46E27

Milestones

Received: 6 February 1975

Revised: 4 March 1977

Published: 1 September 1977