We construct metrizable simplices $X_1$ and $X_2$ and a homeomorphism $\varphi:\overline{ext X_1}\to\overline{ext
X_2}$ such that $\varphi(ext X_1)=ext X_2$, 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. This shows that complementability of
the space $\mathfrak{A}(X)$ cannot be determined by topological properties of the couple $(ext X,\overline{ext X})$.
@article{1222783093,
author = {Ba\v c\'ak, Miroslav and Spurn\'y, Ji\v r\'\i },
title = {Complementability of spaces of affine continuous functions on simplices},
journal = {Bull. Belg. Math. Soc. Simon Stevin},
volume = {15},
number = {1},
year = {2008},
pages = { 465-472},
language = {en},
url = {http://dml.mathdoc.fr/item/1222783093}
}
Bačák, Miroslav; Spurný, Jiří. Complementability of spaces of affine continuous functions on simplices. Bull. Belg. Math. Soc. Simon Stevin, Tome 15 (2008) no. 1, pp. 465-472. http://gdmltest.u-ga.fr/item/1222783093/