Vol. 52, No. 1, 1974

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Isomorphism problems for the Baire classes

Frederick Knowles Dashiell, Jr.

Vol. 52 (1974), No. 1, 29–43

For each ordinal α > 0, the α-th Baire class of bounded measurable functions on the topological space S, denoted α(S), has an algebraic and isometric representation as a space Cα) 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, α(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 ω1(S) of all bounded Baire functions is a C(Ω) space for a certain a-Stonian Ω, α(S) (for α < ω1) is therefore not isomorphic to any complemented subspace of ω1(S).

Mathematical Subject Classification 2000
Primary: 46E15
Secondary: 54C50
Received: 15 August 1973
Published: 1 May 1974
Frederick Knowles Dashiell, Jr.