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October 2017

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Abstract of a paper by T. Domínguez Benavides, M. A, Japón
by Bentuo Zheng (bzheng) 03 Oct '17

03 Oct '17

This is an announcement for the paper “Komlós' Theorem and the Fixed Point Property for affine mappings” by T. Domínguez Benavides<https://arxiv.org/find/math/1/au:+Benavides_T/0/1/0/all/0/1>, M. A<https://arxiv.org/find/math/1/au:+A_M/0/1/0/all/0/1>, Japón<https://arxiv.org/find/math/1/au:+Jap%5C%27on/0/1/0/all/0/1>. Abstract: Assume that $X$ is a Banach space of measurable functions for which Koml\'os' Theorem holds. We associate to any closed convex bounded subset $C$ of $X$ a coefficient $t(C)$which attains its minimum value when $C$ is closed for the topology of convergence in measure and we prove some fixed point results for affine Lipschitzian mappings, depending on the value of $t(C)\in {1, 2]$and the value of the Lipschitz constants of the iterates. As a first consequence, for every $L<2$, we deduce the existence of fixed points for affine uniformly $L$-Lipschitzian mappings defined on the closed unit ball of $L_1([0,1])$. Our main theorem also provides a wide collection of convex closed bounded sets in $L_1([0,1])$ and in some other spaces of functions, which satisfy the fixed point property for affine nonexpansive mappings. Furthermore, this property is still preserved by equivalent renormings when the Banach-Mazur distance is small enough. In particular, we prove that the failure of the fixed point property for affine nonexpansive mappings in $L_1(\mu)$ can only occur in the extremal case $t(C)=2$. Examples are displayed proving that our fixed point theorem is optimal in terms of the Lipschitz constants and the coefficient $t(C)$. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1709.03333

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Abstract of a paper by Elói Medina Galego, Vinícius Morelli Cortes
by Bentuo Zheng (bzheng) 03 Oct '17

03 Oct '17

This is an announcement for the paper “When does $C(K, X)$ contain a complemented copy of $c_0(\Gamma)$ iff $X$ does?” by Elói Medina Galego<https://arxiv.org/find/math/1/au:+Galego_E/0/1/0/all/0/1>, Vinícius Morelli Cortes<https://arxiv.org/find/math/1/au:+Cortes_V/0/1/0/all/0/1>. Abstract: Let $K$ be a compact Hausdorff space with weight $w(K)$, $\tau$ an infinite cardinal with cofinality $cf(\tau)>w(K)$ and $X$ a Banach space. In contrast with a classical theorem of Cembranos and Freniche it is shown that if $cf(\tau)>w(K)$ then the space $C(K, X)$ contains a complemented copy of $c_0(\Gamma)$ if and only if $X$ does. This result is optimal for every infinite cardinal $\tau$, in the sense that it can not be improved by replacing the inequality $cf(\tau)>w(K)$ by another weaker than it. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1709.01114

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