Banach April 2015

banach@mathdept.okstate.edu

1 participants
19 discussions

Abstract of a paper by Spiros A. Argyros and Pavlos Motakis
by alspach＠math.okstate.edu 29 Apr '15

29 Apr '15

This is an announcement for the paper "A dual method of constructing hereditarily indecomposable Banach spaces" by Spiros A. Argyros and Pavlos Motakis. Abstract: A new method of defining hereditarily indecomposable Banach spaces is presented. This method provides a unified approach for constructing reflexive HI spaces and also HI spaces with no reflexive subspace. All the spaces presented here satisfy the property that the composition of any two strictly singular operators is a compact one. This yields the first known example of a Banach space with no reflexive subspace such that every operator has a non-trivial closed invariant subspace. Archive classification: math.FA Mathematics Subject Classification: 46B03, 46B06, 46B25, 46B45, 47A15 Remarks: 41 pages Submitted from: pmotakis(a)central.ntua.gr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1504.01564 or http://arXiv.org/abs/1504.01564

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Abstract of a paper by Raphael Clouatre and Kenneth R. Davidson
by alspach＠math.okstate.edu 29 Apr '15

29 Apr '15

This is an announcement for the paper "The unit ball of the predual of $H^\infty(\mathbb{B}_d)$ has no extreme points" by Raphael Clouatre and Kenneth R. Davidson. Abstract: We identify the exposed points of the unit ball of the dual space of the ball algebra. As a corollary, we show that the predual of $H^\infty(\mathbb{B}_d)$ has no extreme points in its unit ball. Archive classification: math.FA Remarks: 6 pages Submitted from: ottokar_1er(a)hotmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1504.01016 or http://arXiv.org/abs/1504.01016

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Abstract of a paper by Emanuele Casini, Enrico Miglierina, and Lukasz Piasecki
by alspach＠math.okstate.edu 06 Apr '15

06 Apr '15

This is an announcement for the paper "A remark on spaces of affine continuous functions on a simplex" by Emanuele Casini, Enrico Miglierina, and Lukasz Piasecki. Abstract: We present an example of an infinite dimensional separable space of affine continuous functions on a Choquet simplex that does not contain a subspace linearly isometric to $c$. This example disproves a result stated in M. Zippin. On some subspaces of Banach spaces whose duals are $L_1$ spaces. Proc. Amer. Math. Soc. 23, (1969), 378-385. Archive classification: math.FA Mathematics Subject Classification: Primary 46B04, Secondary 46B45, 46B25 Submitted from: enrico.miglierina(a)unicatt.it The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1503.09088 or http://arXiv.org/abs/1503.09088

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Abstract of a paper by Grigory Ivanov
by alspach＠math.okstate.edu 06 Apr '15

06 Apr '15

This is an announcement for the paper "Modulus of supporting convexity and supporting smoothness" by Grigory Ivanov. Abstract: We introduce the moduli of the supporting convexity and the supporting smoothness of the Banach space which characterize the deviation of the unit sphere from an arbitrary supporting hyperplane. We show that the modulus of supporting smoothness, the Banas modulus, and the modulus of smoothness are equivalent at zero, respectively the modulus of supporting convexity is equivalent at zero to the modulus of convexity. We prove a Day-Nordlander type result for these moduli. Archive classification: math.FA Submitted from: grigory.ivanov(a)phystech.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1503.08912 or http://arXiv.org/abs/1503.08912

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Abstract of a paper by Emanuele Casini, Enrico Miglierina, and Lukasz Piasecki
by alspach＠math.okstate.edu 06 Apr '15

06 Apr '15

This is an announcement for the paper "Separable Lindenstrauss spaces whose duals lack the weak$^*$ fixed point property for nonexpansive mappings" by Emanuele Casini, Enrico Miglierina, and Lukasz Piasecki. Abstract: In this paper we study the $w^*$-fixed point property for nonexpansive mappings. First we show that the dual space $X^*$ lacks the $w^*$-fixed point property whenever $X$ contains an isometric copy of the space $c$. Then, the main result of our paper provides several characterizations of weak-star topologies that fail the fixed point property for nonexpansive mappings in $\ell_1$ space. This result allows us to obtain a characterization of all separable Lindenstrauss spaces $X$ inducing the failure of $w^*$-fixed point property in $X^*$. Archive classification: math.FA Mathematics Subject Classification: Primary 47H09, Secondary 46B25 Submitted from: enrico.miglierina(a)unicatt.it The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1503.08875 or http://arXiv.org/abs/1503.08875

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Abstract of a paper by Tomasz Kania and Tomasz Kochanek
by alspach＠math.okstate.edu 06 Apr '15

06 Apr '15

This is an announcement for the paper "Uncountable sets of unit vectors that are separated by more than 1" by Tomasz Kania and Tomasz Kochanek. Abstract: Let $X$ be a Banach space. We study the circumstances under which there exists an uncountable set $\mathcal A\subset X$ of unit vectors such that $\|x-y\|>1$ for distinct $x,y\in \mathcal A$. We prove that such a set exists if $X$ is quasi-reflexive and non-separable; if $X$ is additionally super-reflexive then one can have $\|x-y\|\geqslant 1+\varepsilon$ for some $\varepsilon>0$ that depends only on $X$. If $K$ is a compact, Hausdorff space, then $X=C(K)$ contains such a set of cardinality equal to the density of $X$; this solves a problem left open by S. K. Mercourakis and G. Vassiliadis. Archive classification: math.FA math.MG Remarks: 17 pp Submitted from: tomasz.marcin.kania(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1503.08166 or http://arXiv.org/abs/1503.08166

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Abstract of a paper by George Androulakis and Matthew Ziemke
by alspach＠math.okstate.edu 06 Apr '15

06 Apr '15

This is an announcement for the paper "The closedness of the generator of a semigroup" by George Androulakis and Matthew Ziemke. Abstract: We study semigroups of bounded operators on a Banach space such that the members of the semigroup are continuous with respect to various weak topologies and we give sufficient conditions for the generator of the semigroup to be closed with respect to the topologies involved. The proofs of these results use the Laplace transforms of the semigroup. Thus we first give sufficient conditions for Pettis integrability of vector valued functions with respect to scalar measures. Archive classification: math.FA math-ph math.MP Submitted from: giorgis(a)math.sc.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1503.07472 or http://arXiv.org/abs/1503.07472

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Abstract of a paper by Sanne ter Horst, Miek Messerschmidt, and Andre C.M. Ran
by alspach＠math.okstate.edu 06 Apr '15

06 Apr '15

This is an announcement for the paper "Equivalence after extension for compact operators on Banach spaces" by Sanne ter Horst, Miek Messerschmidt, and Andre C.M. Ran. Abstract: In recent years the coincidence of the operator relations equivalence after extension and Schur coupling was settled for the Hilbert space case, by showing that equivalence after extension implies equivalence after one-sided extension. In this paper we investigate consequences of equivalence after extension for compact Banach space operators. We show that generating the same operator ideal is necessary but not sufficient for two compact operators to be equivalent after extension. In analogy with the necessary and sufficient conditions on the singular values for compact Hilbert space operators that are equivalent after extension, we prove the necessity of similar relationships between the $s$-numbers of two compact Banach space operators that are equivalent after extension, for arbitrary $s$-functions. We investigate equivalence after extension for operators on $\ell^{p}$-spaces. We show that two operators that act on different $\ell^{p}$-spaces cannot be equivalent after one-sided extension. Such operators can still be equivalent after extension, for instance all invertible operators are equivalent after extension, however, if one of the two operators is compact, then they cannot be equivalent after extension. This contrasts the Hilbert space case where equivalence after one-sided extension and equivalence after extension are, in fact, identical relations. Finally, for general Banach spaces $X$ and $Y$, we investigate consequences of an operator on $X$ being equivalent after extension to a compact operator on $Y$. We show that, in this case, a closed finite codimensional subspace of $Y$ must embed into $X$, and that certain general Banach space properties must transfer from $X$ to $Y$. We also show that no operator on $X$ can be equivalent after extension to an operator on $Y$, if $X$ and $Y$ are essentially incomparable Banach spaces. Archive classification: math.FA Mathematics Subject Classification: Primary: 47A05, 47B10 Secondary: 47L20, 46B03 Submitted from: mmesserschmidt(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1503.07350 or http://arXiv.org/abs/1503.07350

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Abstract of a paper by Piotr Koszmider
by alspach＠math.okstate.edu 06 Apr '15

06 Apr '15

This is an announcement for the paper "Uncountable equilateral sets in Banach spaces of the form $C(K)$" by Piotr Koszmider. Abstract: The paper is concerned with the problem whether a nonseparable Banach space must contain an uncountable set of vectors such that the distances between every two distinct vectors of the set are the same. Such sets are called equilateral. We show that Martin's axiom and the negation of the continuum hypothesis imply that every nonseparable Banach space of the form $C(K)$ has an uncountable equilateral set. We also show that one cannot obtain such a result without an additional set-theoretic assumption since we construct an example of nonseparable Banach space of the form $C(K)$ which has no uncountable equilateral set (or equivalently no uncountable $(1+\varepsilon)$-separated set in the unit sphere for any $\varepsilon>0$) making another consistent combinatorial assumption. The compact $K$ is a version of the split interval obtained from a sequence of functions which behave in an anti-Ramsey manner. It remains open if there is an absolute example of a nonseparable Banach space of the form different than $C(K)$ which has no uncountable equilateral set. It follows from the results of S. Mercourakis, G. Vassiliadis that our example has an equivalent renorming in which it has an uncountable equilateral set. It remains open if there are consistent examples which have no uncountable equilateral sets in any equivalent renorming. It follows from the results of S. Todorcevic that it is consistent that every nonseparable Banach space has an equivalent renorming in in which it has an uncountable equilateral set. Archive classification: math.FA math.GN math.LO Submitted from: piotr.math(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1503.06356 or http://arXiv.org/abs/1503.06356

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Banach April 2015