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A metric space $(B,d)$ is said to have the fixed point property (FPP) if every 1-Lipschitz
operator $T:B\to B$ has a fixed point.
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During this talk, our metric space will be the closed unit ball of a Banach space of continuous functions $C(K)$, for $K$ a Hausdorff compact topological space. We are interested in identifying topological properties of the compact set $K$ that are connected to the failure or to the fulfilment of the FPP for the closed unit ball $B$ of $C(K)$.
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This question arises after observing that the two opposite behaviours hold. For instance, if $K$ is the one-point compactification of $\mathbb{N}$, it is easy to check that the closed unit ball of $C(K)$ fails the FPP. In contrast, if $K=\beta\mathbb{N}$, the Stone-Cech compactification of $\mathbb{N}$, the closed unit ball does verify the FPP (extremally disconnection, hyperconvexity and injectivity play their role here).
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Firstly, we will prove that $K$ being a topological $F$-space is a necessary condition for the FPP, which will allow us to dismiss the FPP for different classes of compact spaces.
We will next focus on some particular examples of topological $F$-spaces, in particular, on the remainder of $\beta\mathbb{N}$, that is, $\mathbb{N}^*= \beta\mathbb{N}\setminus\mathbb{N}$ formed by all free ultrafilters.
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While $C(\beta\mathbb{N})$ is isometric to $\ell_\infty$, $C(\mathbb{N}^*)$ is isometric to the quotient Banach space $\ell_\infty/c_0$. A natural question arises: Does the closed unit ball of $C(\mathbb{N}^*)$ or, equivalently, the closed unit ball of $\ell_\infty/c_0$, have the FPP?
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We will prove that Boolean algebras, the Stone representation theorem and Boolean retractions will provide us with an answer to the previous question, under the umbrella of the continuum hypothesis.
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Some open problems will be exposed.
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The results included in this talk will be eventually published in a joint paper together with
Antonio Avilés (Murcia University), Christopher Lennard (Pittsburgh University), Gonzalo Martinez-Fernandez (Murcia University) and Adam Stawski (Pittsburgh University)
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