Banach

banach@mathdept.okstate.edu

2549 discussions

Abstract of a paper by M. A. Lopez and S. Reisner
by Dale Alspach 31 Jul '07

31 Jul '07

This is an announcement for the paper "A note on curves equipartition" by M. A. Lopez and S. Reisner. Abstract: The problem of the existence of an equi-partition of a curve in $\R^n$ has recently been raised in the context of computational geometry. The problem is to show that for a (continuous) curve $\Gamma : [0,1] \to \R^n$ and for any positive integer $N$, there exist points $t_0=0<t_1<...<t_{N-1}<1=t_N$, such that $d(\Gamma(t_{i-1}),\Gamma(t_i))=d(\Gamma(t_{i}),\Gamma(t_{i+1}))$ for all $i=1,...,N$, where $d$ is a metric or even a semi-metric (a weaker notion) on $\R^n$. We show here that the existence of such points, in a much broader context, is a consequence of Brower's fixed point theorem. Archive classification: math.FA math.CA Mathematics Subject Classification: 58C30; 47H10 The source file(s), equipartition.tex: 10551 bytes, is(are) stored in gzipped form as 0707.4296.gz with size 4kb. The corresponding postcript file has gzipped size 46kb. Submitted from: reisner(a)math.haifa.ac.il The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0707.4296 or http://arXiv.org/abs/0707.4296 or by email in unzipped form by transmitting an empty message with subject line uget 0707.4296 or in gzipped form by using subject line get 0707.4296 to: math(a)arXiv.org.

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Abstract of a paper by Junsheng Fang, Don Hadwin, Eric Nordgren, and Junhao Shen
by Dale Alspach 31 Jul '07

31 Jul '07

This is an announcement for the paper "Tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property" by Junsheng Fang, Don Hadwin, Eric Nordgren, and Junhao Shen. Abstract: In this paper we set up a representation theorem for tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property in terms of Ky Fan norms. Examples of tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property include unitarily invariant norms on finite factors (type ${\rm II}\sb 1$ factors and $M_n(\cc)$) and symmetric gauge norms on $L^\infty[0,1]$ and $\cc^n$. As the first application, we obtain that the class of unitarily invariant norms on a type ${\rm II}\sb 1$ factor coincides with the class of symmetric gauge norms on $L^\infty[0,1]$ and von Neumann's classical result~\cite{vN} on unitarily invariant norms on $M_n(\cc)$. As the second application, Ky Fan's dominance theorem~\cite{Fan} is obtained for finite von Neumann algebras satisfying the weak Dixmier property. As the third application, some classical results in non-commutative $L^p$-theory (e.g., non-commutative H$\ddot{\text{o}}$lder's inequality, duality and reflexivity of non-commutative $L^p$-spaces) are obtained for general unitarily invariant norms on finite factors. We also investigate the extreme points of $\NN(\M)$, the convex compact set (in the pointwise weak topology) of normalized unitarily invariant norms (the norm of the identity operator is 1) on a finite factor $\M$. We obtain all extreme points of $\NN(M_2(\cc))$ and many extreme points of $\NN(M_n(\cc))$ ($n\geq 3$). For a type ${\rm II}\sb 1$ factor $\M$, we prove that if $t$ ($0\leq t\leq 1$) is a rational number then the Ky Fan $t$-th norm is an extreme point of $\NN(\M)$. Archive classification: math.OA math.FA Mathematics Subject Classification: 46L10, 46L51 Remarks: 56 pages The source file(s), tracial-gauge-norms.tex: 172272 bytes, is(are) stored in gzipped form as 0707.4239.gz with size 39kb. The corresponding postcript file has gzipped size 244kb. Submitted from: jfang(a)cisunix.unh.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0707.4239 or http://arXiv.org/abs/0707.4239 or by email in unzipped form by transmitting an empty message with subject line uget 0707.4239 or in gzipped form by using subject line get 0707.4239 to: math(a)arXiv.org.

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Abstract of a paper by A.Koldobsky, H.Koenig, and M.Zymonopoulou
by Dale Alspach 31 Jul '07

31 Jul '07

This is an announcement for the paper "The complex Busemann-Petty problem on sections of convex bodies" by A.Koldobsky, H.Koenig, and M.Zymonopoulou. Abstract: The complex Busemann-Petty problem asks whether origin symmetric convex bodies in $\C^n$ with smaller central hyperplane sections necessarily have smaller volume. We prove that the answer is affirmative if $n\le 3$ and negative if $n\ge 4.$ Archive classification: math.FA math.MG Mathematics Subject Classification: 52A20 Remarks: 18 pages The source file(s), complexbp.tex: 46749 bytes, is(are) stored in gzipped form as 0707.3851.gz with size 14kb. The corresponding postcript file has gzipped size 101kb. Submitted from: koldobsk(a)math.missouri.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0707.3851 or http://arXiv.org/abs/0707.3851 or by email in unzipped form by transmitting an empty message with subject line uget 0707.3851 or in gzipped form by using subject line get 0707.3851 to: math(a)arXiv.org.

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Abstract of a paper by Marco Abate and Jean-Pierre Vigue
by Dale Alspach 23 Jul '07

23 Jul '07

This is an announcement for the paper "Isometries for the Carathedory Metric" by Marco Abate and Jean-Pierre Vigue. Abstract: Under certain hypothesises, we prove that a map which is an isometry for the Caratheodory infinitesimal metric at a point is an analytic isomorphism onto its image. Archive classification: math.FA math.CV Mathematics Subject Classification: 32H99 Remarks: 6 pages The source file(s), abate-vigue.tex: 14563 bytes, is(are) stored in gzipped form as 0707.2329.gz with size 5kb. The corresponding postcript file has gzipped size 60kb. Submitted from: vigue(a)math.univ-poitiers.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0707.2329 or http://arXiv.org/abs/0707.2329 or by email in unzipped form by transmitting an empty message with subject line uget 0707.2329 or in gzipped form by using subject line get 0707.2329 to: math(a)arXiv.org.

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Abstract of a paper by Konrad J Swanepoel
by Dale Alspach 23 Jul '07

23 Jul '07

This is an announcement for the paper "Extremal problems in Minkowski space related to minimal networks" by Konrad J Swanepoel. Abstract: We solve the following problem of Z. F\"uredi, J. C. Lagarias and F. Morgan [FLM]: Is there an upper bound polynomial in $n$ for the largest cardinality of a set S of unit vectors in an n-dimensional Minkowski space (or Banach space) such that the sum of any subset has norm less than 1? We prove that |S|\leq 2n and that equality holds iff the space is linearly isometric to \ell^n_\infty, the space with an n-cube as unit ball. We also remark on similar questions raised in [FLM] that arose out of the study of singularities in length-minimizing networks in Minkowski spaces. Archive classification: math.MG math.FA Mathematics Subject Classification: 52A40 (Primary) 52A21, 49Q10 (Secondary) Citation: Proceedings of the American Mathematical Society 124 (1996) The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0707.3052 or http://arXiv.org/abs/0707.3052 or by email in unzipped form by transmitting an empty message with subject line uget 0707.3052 or in gzipped form by using subject line get 0707.3052 to: math(a)arXiv.org.

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Abstract of a paper by T s s R K Rao
by Dale Alspach 23 Jul '07

23 Jul '07

This is an announcement for the paper "Nice surjections on spaces of operators" by T s s R K Rao. Abstract: A bounded linear operator is said to be nice if its adjoint preserves extreme points of the dual unit ball. Motivated by a description due to Labuschagne and Mascioni \cite{LM} of such maps for the space of compact operators on a Hilbert space, in this article we consider a description of nice surjections on ${\mathcal K}(X,Y)$ for Banach spaces $X,Y$. We give necessary and sufficient conditions when nice surjections are given by composition operators. Our results imply automatic continuity of these maps with respect to other topologies on spaces of operators. We also formulate the corresponding result for ${\mathcal L}(X,Y)$ thereby proving an analogue of the result from \cite{LM} for $L^p$ ($1 <p \neq 2 <\infty$) spaces. We also formulate results when nice operators are not of the canonical form, extending and correcting the results from \cite{KS}. Archive classification: math.FA Remarks: 8 pages The source file(s), mat01.cls: 37299 bytes The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0707.2140 or http://arXiv.org/abs/0707.2140 or by email in unzipped form by transmitting an empty message with subject line uget 0707.2140 or in gzipped form by using subject line get 0707.2140 to: math(a)arXiv.org.

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Abstract of a paper by Yongfu Su and Xiaolong Qin
by Dale Alspach 23 Jul '07

23 Jul '07

This is an announcement for the paper "Strong convergence of modified Ishikawa iterations for nonlinear mappings" by Yongfu Su and Xiaolong Qin. Abstract: In this paper, we prove a strong convergence theorem of modified Ishikawa iterations for relatively asymptotically nonexpansive mappings in Banach space. Our results extend and improve the recent results by Nakajo, Takahashi, Kim, Xu, Matsushita and some others. Archive classification: math.FA Mathematics Subject Classification: 47H09, 65J15 Remarks: 11 pages The source file(s), PM2865new.tex: 31156 bytes The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0707.1955 or http://arXiv.org/abs/0707.1955 or by email in unzipped form by transmitting an empty message with subject line uget 0707.1955 or in gzipped form by using subject line get 0707.1955 to: math(a)arXiv.org.

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ANNOUNCEMENT OF SUMIRFAS 2007
by Bill Johnson 12 Jul '07

12 Jul '07

ANNOUNCEMENT OF SUMIRFAS 2007 The Informal Regional Functional Analysis Seminar August 10 - 12 Texas A&M University, College Station Schedule: Talks for SUMIRFAS will be posted on the Workshop in Analysis and Probability page, URL http://www.math.tamu.edu/research/workshops/linanalysis/ Below is a list of speakers, current as of July 6. The first talk will be in the early afternoon on Friday and the Seminar concludes by lunch time on Sunday. All talks will be in Blocker 165. The Blocker Building is on Ireland St. just south of University Dr. on the Texas A&M campus: http://www.tamu.edu/map/building/overview/BLOC.html. Coffee and refreshments will be available in Blocker 155. The usual SUMIRFAS dinner will be on August 11. It will be a BBQ and swim fest at the home of Jan and Bill Johnson. Gideon Schechtman, and Joel Zinn, are organizing a Concentration Week on "Probability Inequalities with Applications to High Dimensional Phenomena" that will take place August 6 - August 10. The first day will be devoted to introductory talks designed to introduce non experts to the subject. We expect to be able to cover housing for most participants from support the National Science Foundation has provided for the Workshop. Preference will be given to participants who do not have other sources of support, such as sponsored research grants. When you ask Cara to book your room, please tell them if you are requesting support. Minorities, women, graduate students, and young researchers are especially encouraged to apply. For logistical support, please contact Cara Barton, cara(a)math.tamu.edu or Jaime Vykukal, jaime(a)math.tamu.edu. For more information on the Workshop itself, please contact William Johnson, johnson(a)math.tamu.edu, David Larson, larson(a)math.tamu.edu, Gilles Pisier, pisier(a)math.tamu.edu, or Joel Zinn, jzinn(a)math.tamu.edu. For information about the Concentration Week on "Probability Inequalities with Applications to High Dimensional Phenomena", please contact Joel Zinn, jzinn(a)math.tamu.edu. SUMIRFAS 2007 Speakers Grahame Bennett, Series of positive terms Paco Garcia, Yehoram Gordon, Best random embedding of $\varepsilon$ nets in convex bodies, and best random $\varepsilon$ Dvoretzky theorem in the $N-$ dimensional cube Adrian Ioana, "Cocycle superrigidity for profinite actions of property (T) groups". Nga Nguyen, Surgery and push-outs on frames Dmitri Panchenko, "Talagrand's positivity principle" Marek Ptak,"Hyperreflexivity of finite-dimensional spaces" Joe Rosenblatt, "Dynamical systems and martingales: the never ending story" Gideon Schechtman, $\ell_p$ strictly singular operators on $L_p$ Staszek Szarek, "Sets of constant height and applications to quantum information theory" Piotr Wojdyllo, "Local commutant approach versus Gabor, Wilson, and wavelet tight frames". Artem Zvavitch, On the local equatorial characterization of zonoids

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Abstract of a paper by David Alonso-Gutierrez
by Dale Alspach 12 Jul '07

12 Jul '07

This is an announcement for the paper "About the isotropy constant of random convex sets" by David Alonso-Gutierrez. Abstract: Let $K$ be the symmetric convex hull of $m$ independent random vectors uniformly distributed on the unit sphere of $\R^n$. We prove that, for every $\delta>0$, the isotropy constant of $K$ is bounded by a constant $c(\delta)$ with high probability, provided that $m\geq (1+\delta)n$. Archive classification: math.FA Mathematics Subject Classification: 52A20; 52A40; 46B20; Remarks: 8 pages The source file(s), Randomconvexsets8.tex: 18946 bytes, is(are) stored in gzipped form as 0707.1570.gz with size 6kb. The corresponding postcript file has gzipped size 72kb. Submitted from: 498220(a)celes.unizar.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0707.1570 or http://arXiv.org/abs/0707.1570 or by email in unzipped form by transmitting an empty message with subject line uget 0707.1570 or in gzipped form by using subject line get 0707.1570 to: math(a)arXiv.org.

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Abstract of a paper by V.Yaskin
by Dale Alspach 12 Jul '07

12 Jul '07

This is an announcement for the paper "On strict inclusions in hierarchies of convex bodies" by V.Yaskin. Abstract: Let $\mathcal I_k$ be the class of convex $k$-intersection bodies in $\mathbb{R}^n$ (in the sense of Koldobsky) and $\mathcal I_k^m$ be the class of convex origin-symmetric bodies all of whose $m$-dimensional central sections are $k$-intersection bodies. We show that 1) $\mathcal I_k^m\not\subset \mathcal I_k^{m+1}$, $k+3\le m<n$, and 2) $\mathcal I_l \not\subset \mathcal I_k$, $1\le k<l < n-3$. Archive classification: math.FA Mathematics Subject Classification: 52A20, 52A21, 46B04 Remarks: 10 pages The source file(s), Yaskin.tex: 31833 bytes, is(are) stored in gzipped form as 0707.1471.gz with size 10kb. The corresponding postcript file has gzipped size 82kb. Submitted from: vyaskin(a)math.ou.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0707.1471 or http://arXiv.org/abs/0707.1471 or by email in unzipped form by transmitting an empty message with subject line uget 0707.1471 or in gzipped form by using subject line get 0707.1471 to: math(a)arXiv.org.

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