Transfinite contractive propagation and the ball fixed point property in \texorpdfstring{C(K)}{C(K)}
Abstract
We prove that, for a compact Hausdorff space K, the real Banach space C(K) has the ball fixed point property if and only if K is extremally disconnected. The new implication is obtained by constructing a fixed-point-free nonexpansive self-map of the whole closed unit ball whenever K is a compact F-space which is not extremally disconnected. The construction uses a transfinite contractive propagation system with two increasing profiles. A successor--limit delay preserves their tail constraint and has no subsolution in the profile domain. Analysis records boundary deficits; positive operators on ordinal C0-spaces and a common center determined by the two tails yield a nonexpansive synthesis satisfying an order domination inequality. The required topological families are obtained from a gap in a maximal Boolean chain in the zero-dimensional case, and from a transfinite extension of signs otherwise. The argument works in ZFC and resolves the question posed by Avilés, Japón, Lennard, Martínez-Cervantes, and Stawski.
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BibTeX
@misc{barroso2026transfinite,
title = {{Transfinite contractive propagation and the ball fixed point property in \texorpdfstring$C(K)$C(K)}},
author = {Cleon S. Barroso},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16318},
eprint = {2609.16318},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16318},
}
RIS
TY - UNPB
TI - Transfinite contractive propagation and the ball fixed point property in \texorpdfstring{C(K)}{C(K)}
AU - Cleon S. Barroso
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.16318
UR - https://arxiv.org/abs/2609.16318
AB - We prove that, for a compact Hausdorff space K, the real Banach space C(K) has the ball fixed point property if and only if K is extremally disconnected. The new implication is obtained by constructing a fixed-point-free nonexpansive self-map of the whole closed unit ball whenever K is a compact F-space which is not extremally disconnected. The construction uses a transfinite contractive propagation system with two increasing profiles. A successor--limit delay preserves their tail constraint and has no subsolution in the profile domain. Analysis records boundary deficits; positive operators on ordinal C_0-spaces and a common center determined by the two tails yield a nonexpansive synthesis satisfying an order domination inequality. The required topological families are obtained from a gap in a maximal Boolean chain in the zero-dimensional case, and from a transfinite extension of signs otherwise. The argument works in ZFC and resolves the question posed by Avilés, Japón, Lennard, Martínez-Cervantes, and Stawski.
ER -