On Å ilov boundary for function spaces

In this paper, we develop the notion of Å ilov boundary for closed, point separating subspaces of spaces of continuous functions on a compact Hausdorff space. We give an example to show that the set of peak points need not be a boundary. We give a proof of Bishop's theorem that describes the Choquet boundary in the case of closed, point separating, subalgebras of continuous functions that do not contain the constant function. For closed point separating subspaces, using results of R.R. Phelps on integral representation theory for subspaces, we show that the closure of the Choquet boundary as a bundle is the Å ilov boundary bundle similar in the sense to the one considered by D. Blecher [5].