Complementability of spaces of affine continuous functions on simplices (Q953983)

The authors construct metrizable simplices \(X_1\) and \(X_2\) and a homeomorphism \(\varphi:\overline{\text{ext } X_1}\to\overline{\text{ext } X_2}\) such that \(\varphi(\text{ext } X_1)=\)ext \(X_2\), the space \(\mathfrak{A}(X_1)\) of all affine continuous functions on \(X_1\) is complemented in \(\mathcal C(X_1)\) and \(\mathfrak{A}(X_2)\) is not complemented in any \(\mathcal C(K)\) space. Thus, the complementability of \(\mathfrak{A}(X)\) cannot be determined from the topological properties of the couple (ext \(X,\overline{\text{ext } X})\).