Banach Spaces of Continuous Functions as Dual Spaces

This is a very specialized monograph in functional analysis, that considers essentially this one question. Suppose \(K\) is a locally compact Hausdorff space. Let \(C_0(K)\) be the space of continuous complex-valued functions on \(K\) that vanish at infinity (that is, for each \(f\) and each \(\epsilon > 0\) there is a compact subset of  \(K\) outside of which \(|f(x)| < \epsilon\)). Then \(C_0(K)\) with the uniform norm is a Banach space. The question considered is: under what conditions on \(K\) is \(C_0(K)\) isomorphic or isometrically isomorphic to the dual (or the bidual) of a Banach space?

This subject goes back about eighty years, and the pioneers were Marshall Stone, Jacques Dixmier, Alexander Grothendieck, and William G. Bade (Bade was the first to present a coherent theory). The present book attempts to present the state of the art in the subject, including some new results of the authors. About the first three-quarters of the develops the necessary background, drawing together material from Banach spaces, Banach lattices, Banach algebras, and \(C^*\) algebras. The remaining one-quarter (the last chapter) states and proves the characterization theorems. The characterization is not complete yet, and the book ends with some open questions.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web page is allenstenger.com. His mathematical interests are number theory and classical analysis.