- Banach - mathdept.okstate.edu

Abstract of a paper by Yanqi Qiu
by alspach＠math.okstate.edu 12 Apr '13

12 Apr '13

This is an announcement for the paper "A remark on the complex interpolation for families of Banach spaces" by Yanqi Qiu. Abstract: We show by explicit examples that the complex interpolation for families of Banach spaces is not stable under rearrangement of the given family on the boundary, although, by well-known results, it is stable when the latter family takes only 2 values. In our examples, we can even assume that the family takes only 3 values, which is best possible. We also characterize all the transformations on the circle that are invariant for complex interpolation at 0, they are precisely the origin-preserving inner functions. Archive classification: math.FA Remarks: 19 pages Submitted from: yqi.qiu(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1304.1403 or http://arXiv.org/abs/1304.1403

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Abstract of a paper by Marek Cuth and Marian Fabian
by alspach＠math.okstate.edu 12 Apr '13

12 Apr '13

This is an announcement for the paper "Projections in duals to Asplund spaces made without Simons' lemma" by Marek Cuth and Marian Fabian. Abstract: G. Godefroy and the second author of this note proved in 1988 that in duals to Asplund spaces there always exists a projectional resolution of the identity. A few years later, Ch. Stegall succeeded to drop from the original proof a deep lemma of S. Simons. Here, we rewrite the condensed argument of Ch. Stegall in a more transparent and detailed way. We actually show that this technology of Ch. Stegall leads to a bit stronger/richer object ---the so called projectional skeleton--- recently constructed by W. Kubi\'s, via S. Simons' lemma and with help of elementary submodels from logic. Archive classification: math.FA Submitted from: cuthm5am(a)karlin.mff.cuni.cz The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1304.1313 or http://arXiv.org/abs/1304.1313

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Abstract of a paper by Mario Chica, Vladimir Kadets, Miguel Martin, Soledad Moreno and Fernando Rambla
by alspach＠math.okstate.edu 12 Apr '13

12 Apr '13

This is an announcement for the paper "The Bishop-Phelps-Bollob\'as moduli of a Banach space" by Mario Chica, Vladimir Kadets, Miguel Martin, Soledad Moreno and Fernando Rambla. Abstract: We introduce two Bishop-Phelps-Bollob\'as moduli of a Banach space which measure, for a given Banach space, what is the best possible Bishop-Phelps-Bollob\'as theorem in this space. We show that there is a common upper bound for these moduli for all Banach spaces and we present an example showing that this bound is sharp. We prove the continuity of these moduli and an inequality with respect to duality. We calculate the two moduli for Hilbert spaces and also present many examples for which the moduli have the maximum possible value (among them, there are $C(K)$ spaces and $L_1(\mu)$ spaces). Finally, we show that if a Banach space has the maximum possible value of any of the moduli, then it contains almost isometric copies of the real space $\ell_\infty^{(2)}$ and present an example showing that this condition is not sufficient. Archive classification: math.FA Mathematics Subject Classification: 46B04 Remarks: 26 pages, 5 figures Submitted from: mmartins(a)ugr.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1304.0376 or http://arXiv.org/abs/1304.0376

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Abstract of a paper by Yun Sung Choi, Sun Kwang Kim, Han Ju Lee and Miguel Martin
by alspach＠math.okstate.edu 12 Apr '13

12 Apr '13

This is an announcement for the paper "The Bishop-Phelps-Bollob\'{a}s theorem for operators on $L_1(\mu)$" by Yun Sung Choi, Sun Kwang Kim, Han Ju Lee and Miguel Martin. Abstract: In this paper we show that the Bishop-Phelps-Bollob\'as theorem holds for $\mathcal{L}(L_1(\mu), L_1(\nu))$ for all measures $\mu$ and $\nu$ and also holds for $\mathcal{L}(L_1(\mu),L_\infty(\nu))$ for every arbitrary measure $\mu$ and every localizable measure $\nu$. Finally, we show that the Bishop-Phelps-Bollob\'as theorem holds for two classes of bounded linear operators from a real $L_1(\mu)$ into a real $C(K)$ if $\mu$ is a finite measure and $K$ is a compact Hausdorff space. In particular, one of the classes includes all Bochner representable operators and all weakly compact operators. Archive classification: math.FA Mathematics Subject Classification: Primary 46B20, Secondary 46B04, 46B22 Submitted from: hanjulee(a)dongguk.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1303.6078 or http://arXiv.org/abs/1303.6078

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Abstract of a paper by Bernhard G. Bodmann
by alspach＠math.okstate.edu 12 Apr '13

12 Apr '13

This is an announcement for the paper "Random fusion frames are nearly equiangular and tight" by Bernhard G. Bodmann. Abstract: This paper demonstrates that random, independently chosen equi-dimensional subspaces with a unitarily invariant distribution in a real Hilbert space provide nearly tight, nearly equiangular fusion frames. The angle between a pair of subspaces is measured in terms of the Hilbert-Schmidt inner product of the corresponding orthogonal projections. If the subspaces are selected at random, then a measure concentration argument shows that these inner products concentrate near an average value. Overwhelming success probability for near tightness and equiangularity is guaranteed if the dimension of the subspaces is sufficiently small compared to that of the Hilbert space and if the dimension of the Hilbert space is small compared to the sum of all subspace dimensions. Archive classification: math.FA Remarks: 12 pages AMS LaTeX, no figures Submitted from: bgb(a)math.uh.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1303.5816 or http://arXiv.org/abs/1303.5816

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Abstract of a paper by S. Patnaik and G. Weiss
by alspach＠math.okstate.edu 12 Apr '13

12 Apr '13

This is an announcement for the paper "A survey on subideals of operators and an introduction to subideal-traces" by S. Patnaik and G. Weiss. Abstract: Operator ideals in B(H) are well understood and exploited but ideals inside them have only recently been studied starting with the 1983 seminal work of Fong and Radjavi and continuing with two recent articles by the authors of this survey. This article surveys this study embodied in these three articles. A subideal is a two-sided ideal of J (for specificity also called a J-ideal) for J an arbitrary ideal of B(H). In this terminology we alternatively call J a B(H)-ideal. This surveys these three articles in which we developed a complete characterization of all J-ideals generated by sets of cardinality strictly less than the cardinality of the continuum. So a central theme is the impact of generating sets for subideals on their algebraic structure. This characterization includes in particular finitely and countably generated J-ideals. It was obtained by first generalizing to arbitrary principal J-ideals the 1983 work of Fong-Radjavi who determined which principal K(H)-ideals are also B(H)-ideals. A key property in our investigation turned out to be J-softness of a B(H)-ideal I inside J, that is, IJ = I, a generalization of a recent notion of K(H)-softness of B(H)-ideals introduced by Kaftal-Weiss and earlier exploited for Banach spaces by Mityagin and Pietsch. This study of subideals and the study of elementary operators with coefficient constraints are closely related. Here we also introduce and study a notion of subideal-traces where classical traces (unitarily invariant linear functionals) need not make sense for subideals that are not B(H)-ideals. Archive classification: math.OA math.FA Mathematics Subject Classification: Primary: 47L20, 47B10, 47B07, Secondary: 47B47, 47B37, 13C05, Remarks: 9 pages preprint Submitted from: patnaisa(a)mail.uc.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1303.5697 or http://arXiv.org/abs/1303.5697

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Abstract of a paper by Gero Fendler and Michael Leinert
by alspach＠math.okstate.edu 12 Apr '13

12 Apr '13

This is an announcement for the paper "Separable $C^{\ast}$-algebras and weak$^{\ast}$-fixed point property" by Gero Fendler and Michael Leinert. Abstract: It is shown that the dual $\widehat{A}$ of a separable $C^{\ast}$-algebra $A$ is discrete if and only if its Banach space dual has the weak$^{\ast}$-fixed point property. We prove further that these properties are equivalent to the uniform weak$^{\ast}$ Kadec-Klee property of $A^{\ast}$ and to the coincidence of the weak$^{\ast}$ topology with the norm topology on the pure states of $A$. Archive classification: math.OA Mathematics Subject Classification: Primary: 46L05, 47L50, Secondary: 46L30, 47H10 Remarks: 6 pages Submitted from: gero.fendler(a)univie.ac.at The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1303.5557 or http://arXiv.org/abs/1303.5557

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Abstract of a paper by David Alonso-Gutierrez and Joscha Prochno
by alspach＠math.okstate.edu 12 Apr '13

12 Apr '13

This is an announcement for the paper "Mean width of random perturbations of random polytopes" by David Alonso-Gutierrez and Joscha Prochno. Abstract: We prove some "high probability" results on the expected value of the mean width for random perturbations of random polytopes. The random perturbations are considered for Gaussian and $p$-stable random vectors, as well as uniform distributions on $\ell_p^N$-balls and the unit sphere. Archive classification: math.FA math.PR Mathematics Subject Classification: Primary 52A22, Secondary 52A23, 05D40 Submitted from: joscha.prochno(a)jku.at The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1303.5677 or http://arXiv.org/abs/1303.5677

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Abstract of a paper by P. Wojtaszczyk
by alspach＠math.okstate.edu 12 Apr '13

12 Apr '13

This is an announcement for the paper "On left democracy function" by P. Wojtaszczyk. Abstract: We continue the study undertaken in \cite{GHN} of left democracy function $h_l(N)=\inf_{\#\Lambda=N}\left\|\sum_{n\in \Lambda_N} x_n\right\| $ of an unconditional basis in a Banach space $X$. We provide an example of a basis with $h_l$ non-doubling. Then we show that for bases with non-doubling $h_l$ the greedy projection is not optimal. Together with results from \cite{GHN} and improved by C. Cabrelli, G. Garrig\'os, E. Hernandez and U. Molter we get that the basis is greedy if and only if the greedy projection is optimal. Archive classification: math.FA Submitted from: wojtaszczyk(a)mimuw.edu.pl The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1303.4972 or http://arXiv.org/abs/1303.4972

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Abstract of a paper by Daniel Beltita, Sasmita Patnaik, and Gary Weiss
by alspach＠math.okstate.edu 12 Apr '13

12 Apr '13

This is an announcement for the paper "$B(H)$-Commutators: A historical survey II and recent advances on commutators of compact operators" by Daniel Beltita, Sasmita Patnaik, and Gary Weiss. Abstract: A sequel to \cite{gW05}, we address again the single commutator problem \cite{PT71} of Pearcy and Topping: Is every compact operator a single commutator of compact operators? by focusing on a 35 year old test question for this posed in 1976 by the last named author and others: Are there any strictly positive operators that are single commutators of compact operators? The latter we settle here affirmatively with a modest modification of Anderson's fundamental construction \cite{jA77} constructing compact operators whose commutator is a rank one projection. Moreover we provide here a rich class of such strictly positive operators that are commutators of compact operators and pose a question for the rest. We explain also how these methods are related to the study of staircase matrix forms, their equivalent block tri-diagonal forms, and commutator problems. In particular, we present the original test question and solution that led to the negative solution of the Pearcy-Topping question on whether or not every trace class trace zero operator was a commutator (or linear combination of commutators) of Hilbert-Schmidt operators. And we show how this evolved from staircase form considerations along with a Larry Brown result on trace connections to ideals \cite{lB94} which itself is at the core of \cite[Section 7]{DFWW}. The omission in \cite{gW05} of this important 35 year old test question was inadvertent and we correct that in this paper. This sequel starts where [ibid] left off but can be read independently of [ibid]. The present paper also has a section on self-commutator equations $[X^*,X]=A$ within the framework of some classical operator Lie algebras. That problem was solved by Fan and Fong (1980) for the full algebra of compact operators, and we solve it here for the complex symplectic Lie algebra of compact operators and for complex semisimple Lie algebras. Archive classification: math.OA math.FA math.RT Mathematics Subject Classification: Primary: 47B47, 47B10, 47L20, Secondary: 47-02, 47L30, 17B65, Remarks: 20 pages Submitted from: Daniel.Beltita(a)imar.ro The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1303.4844 or http://arXiv.org/abs/1303.4844

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