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May 2018

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Abstract of a paper by Rainis Haller, Johann Langemets, Vegard Lima, Rihhard Nadel
by Bentuo Zheng (bzheng) 05 May '18

05 May '18

This is an announcement for the paper “Symmetric strong diameter two property” by Rainis Haller<https://arxiv.org/search?searchtype=author&query=Haller%2C+R>, Johann Langemets<https://arxiv.org/search?searchtype=author&query=Langemets%2C+J>, Vegard Lima<https://arxiv.org/search?searchtype=author&query=Lima%2C+V>, Rihhard Nadel<https://arxiv.org/search?searchtype=author&query=Nadel%2C+R>. Abstract: We study Banach spaces with the property that, given a finite number of slices of the unit ball, there exists a direction such that all these slices contain a line segment of length almost 2 in this direction. This property was recently named the symmetric strong diameter two property by Abrahamsen, Nygaard, and P\~oldvere. The symmetric strong diameter two property is not just formally stronger than the strong diameter two property (finite convex combinations of slices have diameter 2. We show that the symmetric strong diameter two property is only preserved by $\ell_{\infty}$-sums, and working with weak star slices we show that $\Lip_0(M)$ have the weak star version of the property for several classes of metric spaces $M$. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1804.01705

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Abstract of a paper by Richard Lechner
by Bentuo Zheng (bzheng) 05 May '18

05 May '18

This is an announcement for the paper “Subsymmetric weak$^*$ Schauder bases and factorization of the identity” by Richard Lechner<https://arxiv.org/find/math/1/au:+Lechner_R/0/1/0/all/0/1>. Abstract: Let $X^*$ denote a Banach space with a subsymmetric weak$^*$ Schauder basis satisfying condition. We show that for any operator $T: X^*\rightarrow X^*$ either $T(X^*)$ or $(I-T)(X^*)$ contains a subspace that is isomorphic to $X^*$ and complemented in $X^*$. Moreover, we prove that $\ell_p(X^*), 1\leq p\leq\infty$ is primary. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1804.01372

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