Banach December 2011

banach@mathdept.okstate.edu

2 participants
34 discussions

Abstract of a paper by Joanna Garbulinska and Wiesaw Kubis
by alspach＠math.okstate.edu 20 Dec '11

20 Dec '11

This is an announcement for the paper "Remarks on Gurarii spaces" by Joanna Garbulinska and Wiesaw Kubis. Abstract: We present selected known results and some of their improvements, involving Gurarii spaces. A Banach space is Gurarii if it has certain natural extension property for almost isometric embeddings of finite-dimensional spaces. Deleting the word ``almost", we get the notion of a strong Gurarii space. There exists a unique (up to isometry) separable Gurarii space, however strong Gurarii spaces cannot be separable. The structure of the class of non-separable Gurarii spaces seems to be not very well understood. We discuss some of their properties and state some open questions. In particular, we characterize non-separable Gurarii spaces in terms of skeletons of separable subspaces, we construct a non-separable Gurarii space with a projectional resolution of the identity and we show that no strong Gurarii space can be weakly Lindel\"of determined. Archive classification: math.FA Mathematics Subject Classification: 46B04, 46B20 Remarks: 30 pages Submitted from: kubis(a)math.cas.cz The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1111.5840 or http://arXiv.org/abs/1111.5840

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Abstract of a paper by Frank Morgan and Aldo Pratelli
by alspach＠math.okstate.edu 20 Dec '11

20 Dec '11

This is an announcement for the paper "Existence of isoperimetric regions in $\R^n$ with density" by Frank Morgan and Aldo Pratelli. Abstract: We prove the existence of isoperimetric regions in $\R^n$ with density under various hypotheses on the growth of the density. Along the way we prove results on the boundedness of isoperimetric regions. Archive classification: math.FA math.AP Remarks: 31 pages, 4 figures Submitted from: aldo.pratelli(a)unipv.it The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1111.5160 or http://arXiv.org/abs/1111.5160

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Abstract of a paper by J. Lopez-Abad and S. Todorcevic
by alspach＠math.okstate.edu 20 Dec '11

20 Dec '11

This is an announcement for the paper "Positional graphs and conditional structure of weakly null sequences" by J. Lopez-Abad and S. Todorcevic. Abstract: We prove that, unless assuming additional set theoretical axioms, there are no reflexive space without unconditional sequences of density the continuum. We give for every integer $n$ there are normalized weakly-null sequences of length $\om_n$ without unconditional subsequences. This together with a result of \cite{Do-Lo-To} shows that $\om_\omega$ is the minimal cardinal $\kappa$ that could possibly have the property that every weakly null $\kappa$-sequence has an infinite unconditional basic subsequence . We also prove that for every cardinal number $\ka$ which is smaller than the first $\om$-Erd\"os cardinal there is a normalized weakly-null sequence without subsymmetric subsequences. Finally, we prove that mixed Tsirelson spaces of uncountable densities must always contain isomorphic copies of either $c_0$ or $\ell_p$, with $p\ge 1$. Archive classification: math.FA math.LO Mathematics Subject Classification: Primary 46B03, 03E35, Secondary 03E02, 03E55, 46B26, 46A35 Submitted from: abad(a)icmat.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1111.5150 or http://arXiv.org/abs/1111.5150

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Abstract of a paper by Spiros A. Argyros and Kevin Beanland
by alspach＠math.okstate.edu 20 Dec '11

20 Dec '11

This is an announcement for the paper "On spaces admitting no $\ell_p$ or $c_0$ spreading model" by Spiros A. Argyros and Kevin Beanland. Abstract: It is shown that for each separable Banach space $X$ not admitting $\ell_1$ as a spreading model there is a space $Y$ having $X$ as a quotient and not admitting any $\ell_p$ for $1 \leq p < \infty$ or $c_0$ as a spreading model. We also include the solution to a question of W.B. Johnson and H.P. Rosenthal on the existence of a separable space not admitting as a quotient any space with separable dual. Archive classification: math.FA Mathematics Subject Classification: 46B06 Remarks: 17 pages Submitted from: kbeanland(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1111.4714 or http://arXiv.org/abs/1111.4714

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Banach December 2011