For each ordinal α > 0, the α-th
Baire class of bounded measurable functions on the topological space S, denoted
ℬα(S), has an algebraic and isometric representation as a space C(Ωα) of
all continuous functions on a totally disconnected compact space Ωα. This
representation is used to study the Baire classes from the point of view of
nonseparable Banach spaces of continuous functions. It is shown that if the compact
space S contains an uncountable compact metrizable subset, then, for each countable
ordinal α, ℬα(S) is not isomorphic (i.e., linearly homeomorphic) to any
complemented subspace of a Banach space C(Ω) for a-Stonian Ω. Since the space
ℬω1(S) of all bounded Baire functions is a C(Ω) space for a certain a-Stonian Ω,
ℬα(S) (for α < ω1) is therefore not isomorphic to any complemented subspace of
ℬω1(S).
