Bulletin of the Belgian Mathematical Society - Simon Stevin

Complementability of spaces of affine continuous functions on simplices

Miroslav Bačák and Jiří Spurný

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Abstract

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 information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 15, Number 3 (2008), 465-472.

Dates
First available in Project Euclid: 30 September 2008

Permanent link to this document
https://projecteuclid.org/euclid.bbms/1222783093

Digital Object Identifier
doi:10.36045/bbms/1222783093

Mathematical Reviews number (MathSciNet)
MR2457962

Zentralblatt MATH identifier
1159.46005

Subjects
Primary: 46A55: Convex sets in topological linear spaces; Choquet theory [See also 52A07]
Secondary: 46B03: Isomorphic theory (including renorming) of Banach spaces

Keywords
simplex $L^1$--predual complementability affine functions

Citation

Bačák, Miroslav; Spurný, Jiří. Complementability of spaces of affine continuous functions on simplices. Bull. Belg. Math. Soc. Simon Stevin 15 (2008), no. 3, 465--472. doi:10.36045/bbms/1222783093. https://projecteuclid.org/euclid.bbms/1222783093


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