Complementability of spaces of affine continuous functions on simplices. (English) Zbl 1159.46005

Bačák, Miroslav; Spurný, Jiří

Bull. Belg. Math. Soc. - Simon Stevin 15, No. 3, 465-472 (2008).

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 \(X,\overline{\text{ext } X})\).

Reviewer: S. S. Kutateladze (Novosibirsk)

MSC:

46A55	Convex sets in topological linear spaces; Choquet theory
46B03	Isomorphic theory (including renorming) of Banach spaces
46B25	Classical Banach spaces in the general theory

Keywords:

simplex; \(L^1\)-predual; complementability; affine function