Linear extension operators on products of compact spaces (Q1403815)

Given a dense subspace \(X\) of a space \(Z\), put \(\eta(X,Z)=\) inf\(\{\|T\|:T\) is a linear extension operator from \(C^*(R(X,Z))\) to \(C^*(Z)\}\), here \(R(X,Z)\) is the remainder \(Z\setminus X\). It is observed that in general \(1\leq\eta(X\times Y,\widehat X\times\widehat Y)\leq 3\), where \(\widehat X\) is the Alexandroff compactification of the locally compact space \(X\). The goal of this paper is to find conditions on \(X\) and \(Y\) which characterize the cases \(\eta(X\times Y,\widehat X\times\widehat Y)\geq 2\) and \(\eta(X\times Y,\widehat X\times\widehat Y)<2\). One of the main results is that the inequality \(\eta(X\times Y,\widehat X\times\widehat Y)<3\) implies the existence of a neighborhood base of the points \(\infty_X\) and \(\infty_Y\), linearly ordered by inclusion and with the same cardinal number. From this fact, it is shown that \(X\) and \(Y\) are either simultaneously pesudocompact or simultaneously nonpseudocompact.