This paper initiates a study of
the classes of Baire measurable functions on the unit interval I from the standpoint
of the theory of spaces of continuous functions. For each countable ordinal α, the
α-th Baire class Bα has a representation as C(Ωα), where Ωα is a certain
compactification of the discrete set I. For 1 ≦ α < β,Bα is a closed subalgebra of Bβ.
The principal result proved here is the fact that Bα is always uncomplemented as a
closed subspace of Bβ. The method of proof relies on a detailed analysis on the
canonical onto map ϕ : Ωβ→ Ωα induced by the imbedding of Bα in Bβ, and
consists of showing that this map admits no “averaging operator.” It depends
heavily on recent results in the theory of averaging operators due to S. Z.
Ditor.
