December 2016 - Banach - mathdept.okstate.edu

Abstract of a paper by Emanuele Casini, Enrico Miglierina, Łukasz Piasecki, Roxana Popescu
by Bentuo Zheng (bzheng) 01 Dec '16

01 Dec '16

This is an announcement for the paper “Stability constants of the weak$^*$ fixed point property for the space $\ell_1$” by Emanuele Casini<https://arxiv.org/find/math/1/au:+Casini_E/0/1/0/all/0/1>, Enrico Miglierina<https://arxiv.org/find/math/1/au:+Miglierina_E/0/1/0/all/0/1>, Łukasz Piasecki<https://arxiv.org/find/math/1/au:+Piasecki_L/0/1/0/all/0/1>, Roxana Popescu<https://arxiv.org/find/math/1/au:+Popescu_R/0/1/0/all/0/1>. Abstract: The main aim of the paper is to study some quantitative aspects of the stability of the weak∗ fixed point property for nonexpansive maps in $\ell_1$ (shortly, $w^*$-fpp). We focus on two complementary approaches to this topic. First, given a predual $X$ of $\ell_1$ such that the $\sigma(\ell_1, X)$-fpp holds, we precisely establish how far, with respect to the Banach-Mazur distance, we can move from X without losing the $w^*$-fpp. The interesting point to note here is that our estimate depends only on the smallest radius of the ball in $\ell_1$ containing all $\sigma(\ell_1, X)$-cluster points of the extreme points of the unit ball. Second, we pass to consider the stability of the $w^*$-fpp in the restricted framework of preduals of $\ell_1$. Namely, we show that every predual $X$ of $\ell_1$ with a distance from $c_0$ strictly less than 3, induces a weak∗ topology on $\ell_1$ such that the $\sigma(\ell_1, X)$-fpp holds. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1611.02133

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Abstract of a paper by Nik Weaver
by Bentuo Zheng (bzheng) 01 Dec '16

01 Dec '16

This is an announcement for the paper “On the unique predual problem for Lipschitz spaces” by Nik Weaver<https://arxiv.org/find/math/1/au:+Weaver_N/0/1/0/all/0/1>. Abstract: For any metric space $X$, the predual of Lip$(X)$ is unique. If $X$ has finite diameter or is complete and convex --- in particular, if it is a Banach space --- then the predual of Lip$_0(X)$ is unique. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1611.01812

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Abstract of a paper by Richard Lechner
by Bentuo Zheng (bzheng) 01 Dec '16

01 Dec '16

This is an announcement for the paper “Factorization in $SL_{\infty}$” by Richard Lechner<https://arxiv.org/find/math/1/au:+Lechner_R/0/1/0/all/0/1>. Abstract: We show that the non-separable Banach space $SL_{\infty}$ is primary. This is achieved by directly solving the infinite dimensional factorization problem in $SL_{\infty}$. In particular, we bypass Bourgain's localization method. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1611.00622

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