Linear extensions of Baire-alpha functions

Abstract

In this paper, we extend the main result on linear extensions of some Baire-one functions, obtained recently by W. Sieg, to a larger class of Baire functions.Let A be a nonempty subset of a Hausdorff space X, and let α be an ordinal number <ω1. By Bα(X) we denote the space of all real functions X→R of Baire-class-α. F(A) is the set of functions A→R with a property F, such that F(A) is a linear subspace of Bα(A). We prove Borsuk–Dugundji-type extension theorems: we give an explicit form of a linear extension operator T:F(A)→F(X), where A is an Fσ and Gδ-subset of a normal space X. Our results apply for F=to be piecewise continuous, and F=to be of Baire-class-α. We show that T restricted to the subspace of bounded functions from F(A) is a positive isometry with the supremum norm. In particular, this solves partially an extension problem set in 2005 by Kalenda and Spurný.The same technique of proof can be used to obtain similar results for functions with values in separable normed spaces.