Complementation problems for the Baire classes

Abstract

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 a, the th Baire class 3B has a representation as C( a ), where a is a certain compactification of the discrete set I. For 1 S a < , 55 rt is a closed subalgebra of $8. The principal result proved here is the fact that 33 is always uncomplemented as a closed subspace of 33/3. The method of proof relies on a detailed analysis on the canonical onto map : -> a induced by the imbedding of 53 in 23, 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.