On linear extension operators from growths of compactifications of products (Q1924682)

Let \(X\) and \(Y\) be locally compact, noncompact spaces. The main theorem in this paper says that if \(\widehat{X}\) and \(\widehat{Y}\) are the one-point-compactifications of \(X\) and \(Y\), and one of \(X\) or \(Y\) is pseudocompact, then every linear extension operator from \(C((\widehat{X} \times \widehat{Y}) \smallsetminus (X\times Y))\) to \(C(\widehat{X} \times \widehat{Y})\) must have norm greater than or equal to two (where \(C(Z)\) denotes the algebra of all real-valued continuous functions on \(Z\)). An example is given of such an operator having norm equal to two. This result extends to any compactification \(K\) of \(X\times Y\) greater than \(\widehat{X} \times \widehat{Y}\), and has as a consequence that \(K\smallsetminus (X\times Y)\) is not a neighborhood retract of \(K\) whenever one of \(X\) or \(Y\) is pseudocompact. The proof of the main theorem uses the dual space of \(C(\widehat{X} \times \widehat{Y})\) consisting of the finite regular Borel measures on \(\widehat{X} \times \widehat{Y}\) with the weak-star topology.