Vol. 72, No. 1, 1977

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The second dual of C(X)

Christine Ann Shannon

Vol. 72 (1977), No. 1, 237–253

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
Received: 6 February 1975
Revised: 4 March 1977
Published: 1 September 1977
Christine Ann Shannon