超限压缩传播与$C(K)$中的球不动点性质

中文摘要 AI 辅助

我们证明，对于紧豪斯多夫空间$K$，实巴拿赫空间$C(K)$具有球不动点性质当且仅当$K$是极不连通的。新的蕴含关系是通过在$K$为紧$F$-空间但非极不连通时，构造整个闭单位球上的无不动点非扩张自映射而获得的。该构造使用具有两个递增轮廓的超限压缩传播系统。后继-极限延迟保持其尾部约束，且在轮廓域中无子解。分析记录边界缺陷；序数$C_0$-空间上的正算子以及由两个尾部确定的公共中心产生满足序支配不等式的非扩张综合。所需的拓扑族在零维情形下由极大布尔链中的间隙获得，在其他情形下由符号的超限扩展获得。该论证在ZFC中成立，并解决了Avilés、Japón、Lennard、Martínez-Cervantes和Stawski提出的问题。

英文摘要

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.

发表机构

• Federal University of Ceará（塞阿拉联邦大学）

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