Differences of bounded semi-continuous functions, I

TL;DR

This paper explores the structural properties of the Banach algebra of differences of semi-continuous functions on metric spaces, providing new characterizations, extension results, and solutions to open problems in the field.

Contribution

It introduces intrinsic characterizations of $SD(K)$, proves new properties of functions in $D(K)$, and solves an open problem regarding the variable oscillation criterion.

Findings

Characterization of functions in $SD(K)$ using transfinite oscillation sets

Extension of real-valued functions from subsets to $D(K)$

Solution to the open problem on the variable oscillation criterion

Abstract

Structural properties are given for $D(K)$, the Banach algebra of (complex) differences of bounded semi-continuous functons on a metric space $K$. For example, it is proved that if all finite derived sets of $K$ are non-empty, then a complex function $\varphi$ operates on $D(K)$ (i.e., $\varphi\circ f\in D(K)$ for all $f\in D(K)$) if and only if $\varphi$ is locally Lipschitz. Another example: if $W\subset K$ and $g\in D(W)$ is real-valued, then it is proved that $g$ extends to a $\tilde g$ in $D(K)$ with $\|\tilde g\|_{D(K)} = \|g\|_{D(W)}$. Considerable attention is devoted to $SD(K)$, the closure in $D(K)$ of the set of simple functions in $D(K)$. Thus it is proved that every member of $SD(K)$ is a (complex) difference of semi-continuous functions in $SD(K)$, and that $|f|$ belongs to $SD(K)$ if $f$ does. An intrinsic characterization of $SD(K)$ is given, in terms of transfinite oscillation sets. Using the transfinite oscillations, alternate proofs are given of the results of Chaatit, Mascioni and Rosenthal that functions of finite Baire-index belong to $SD(K)$, and that $SD(K)\ne D(K)$ for interesting $K$. It is proved that the ``variable oscillation criterion'' characterizes functions belonging to $B_{1/4}(K)$, thus answering an open problem raised in earlier work of Haydon, Odell and Rosenthal. It is also proved that $f$ belongs to $B_{1/4}(K)$ (if and) only if $f$ is a uniform limit of simple $D$-functions of uniformly bounded $D$-norm iff $\osc_\omega f$ is bounded; the last equivalence has also been obtained by V.~Farmaki, using other methods.