Banach February 2005

banach@mathdept.okstate.edu

1 participants
7 discussions

Abstract of a paper by Hun Hee Lee
by Dale Alspach 17 Feb '05

17 Feb '05

This is an announcement for the paper "Weak OH-type 2 and weak OH-cotype 2 of operator spaces" by Hun Hee Lee. Abstract: Recently, OH-type and OH-cotype of operator spaces, an operator space version of type and cotype, were introduced and investigated by the author. In this paper we define weak OH-type 2 (resp. weak OH-cotype 2) of operator spaces, which lies strictly between OH-type 2 (resp. OH-cotype 2) and OH-type $p$ for all $1 \leq p < 2$. (resp. OH-cotype $q$ for all $2< q <= \infty$) This is an analogue of weak type 2 and weak cotype 2 in Banach space case, so we develop analogous theory focusing on the local properties of spaces with such conditions. Archive classification: Functional Analysis; Operator Algebras Remarks: 21 pages The source file(s), WeakOH.tex: 55124 bytes, is(are) stored in gzipped form as 0502337.gz with size 15kb. The corresponding postcript file has gzipped size 85kb. Submitted from: hunmada(a)hanmail.net The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0502337 or http://arXiv.org/abs/math.FA/0502337 or by email in unzipped form by transmitting an empty message with subject line uget 0502337 or in gzipped form by using subject line get 0502337 to: math(a)arXiv.org.

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Abstract of a paper by Hun Hee Lee
by Dale Alspach 16 Feb '05

16 Feb '05

This is an announcement for the paper "Eigenvalues of completely nuclear maps and completely bounded projection constants" by Hun Hee Lee. Abstract: We investigate the distribution of eigenvalues of completely nuclear maps on an operator space. We prove that eigenvalues of completely nuclear maps are square-summable in general and summable if the underlying operator space is Hilbertian and homogeneous. Conversely, if eigenvalues are summable for all completely nuclear maps, then every finite dimensional subspace of the underlying operator space is uniformly completely complemented. As an application we consider an estimate of completely bounded projection constants of $n$-dimensional operator spaces. Archive classification: Functional Analysis; Operator Algebras Remarks: 10 pages The source file(s), EigenComNuclear.tex: 27465 bytes, is(are) stored in gzipped form as 0502335.gz with size 9kb. The corresponding postcript file has gzipped size 55kb. Submitted from: hunmada(a)hanmail.net The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0502335 or http://arXiv.org/abs/math.FA/0502335 or by email in unzipped form by transmitting an empty message with subject line uget 0502335 or in gzipped form by using subject line get 0502335 to: math(a)arXiv.org.

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Abstract of a paper by Hun Hee Lee
by Dale Alspach 16 Feb '05

16 Feb '05

This is an announcement for the paper "OH-type and OH-cotype of operator spaces and completely summing maps" by Hun Hee Lee. Abstract: The definition and basic properties of OH-type and OH-cotype of operator spaces are given. We prove that every bounded linear map from C(K) into OH-cotype q (2<= q < infinity) space (including most of commutative L_q-spaces) for a compact set K satisfies completely (q,2)-summing property, a noncommutative analogue of absolutely (q,2)-summing property. At the end of this paper, we observe that ``OH-cotype 2" is equivalent to the previous definition of ``OH-cotype 2" of G. Pisier. Archive classification: Functional Analysis; Operator Algebras Remarks: 17 pages The source file(s), OH-typecotype.tex: 46265 bytes, is(are) stored in gzipped form as 0502302.gz with size 13kb. The corresponding postcript file has gzipped size 80kb. Submitted from: hunmada(a)hanmail.net The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0502302 or http://arXiv.org/abs/math.FA/0502302 or by email in unzipped form by transmitting an empty message with subject line uget 0502302 or in gzipped form by using subject line get 0502302 to: math(a)arXiv.org.

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Abstract of a paper by Mark Rudelson and Roman Vershynin
by Dale Alspach 16 Feb '05

16 Feb '05

This is an announcement for the paper "Geometric approach to error correcting codes and reconstruction of signals" by Mark Rudelson and Roman Vershynin. Abstract: We develop an approach through geometric functional analysis to error correcting codes and to reconstruction of signals from few linear measurements. An error correcting code encodes an n-letter word x into an m-letter word y in such a way that x can be decoded correctly when any r letters of y are corrupted. We prove that most linear orthogonal transformations Q from R^n into R^m form efficient and robust robust error correcting codes over reals. The decoder (which corrects the corrupted components of y) is the metric projection onto the range of Q in the L_1 norm. An equivalent problem arises in signal processing: how to reconstruct a signal that belongs to a small class from few linear measurements? We prove that for most sets of Gaussian measurements, all signals of small support can be exactly reconstructed by the L_1 norm minimization. This is a substantial improvement of recent results of Donoho and of Candes and Tao. An equivalent problem in combinatorial geometry is the existence of a polytope with fixed number of facets and maximal number of lower-dimensional facets. We prove that most sections of the cube form such polytopes. Archive classification: Functional Analysis; Combinatorics Mathematics Subject Classification: 46B07; 94B75, 68P30, 52B05 Remarks: 17 pages, 3 figures The source file(s), ecc.tex: 50560 bytes, ecc1.eps: 4526 bytes, ecc2.eps: 17097 bytes, ecc3.eps: 4645 bytes, is(are) stored in gzipped form as 0502299.tar.gz with size 23kb. The corresponding postcript file has gzipped size 84kb. Submitted from: vershynin(a)math.ucdavis.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0502299 or http://arXiv.org/abs/math.FA/0502299 or by email in unzipped form by transmitting an empty message with subject line uget 0502299 or in gzipped form by using subject line get 0502299 to: math(a)arXiv.org.

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Abstract of a paper by Jesus M. F. Castillo, Yolanda Moreno and Jesus Suarez
by Dale Alspach 10 Feb '05

10 Feb '05

This is an announcement for the paper "On Lindenstrauss-Pelczynski spaces" by Jesus M. F. Castillo, Yolanda Moreno and Jesus Suarez. Abstract: In this work we shall be concerned with some stability aspects of the classical problem of extension of $C(K)$-valued operators. We introduce the class $\mathscr{LP}$ of Banach spaces of Lindenstrauss-Pe\l czy\'{n}sky type as those such that every operator from a subspace of $c_0$ into them can be extended to $c_0$. We show that all $\mathscr{LP}$-spaces are of type $\mathcal L_\infty$ but not the converse. Moreover, $\mathcal L_\infty$-spaces will be characterized as those spaces $E$ such that $E$-valued operators from $w^*(l_1,c_0)$-closed subspaces of $l_1$ extend to $l_1$. Complemented subspaces of $C(K)$ and separably injective spaces are subclasses of $\mathscr{LP}$-spaces and we show that the former does not contain the latter. It is established that $\mathcal L_\infty$-spaces not containing $l_1$ are quotients of $\mathscr{LP}$-spaces, while $\mathcal L_\infty$-spaces not containing $c_0$, quotients of an $\mathscr{LP}$-space by a separably injective space and twisted sums of $\mathscr{LP}$-spaces are $\mathscr{LP}$-spaces. Archive classification: Functional Analysis Mathematics Subject Classification: 46B03; 46M99; 46B07 The source file(s), CastilloMorenoLP.tex: 49873 bytes, is(are) stored in gzipped form as 0502081.gz with size 15kb. The corresponding postcript file has gzipped size 72kb. Submitted from: castillo(a)unex.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0502081 or http://arXiv.org/abs/math.FA/0502081 or by email in unzipped form by transmitting an empty message with subject line uget 0502081 or in gzipped form by using subject line get 0502081 to: math(a)arXiv.org.

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Abstract of a paper by Valentin Ferenczi
by Dale Alspach 04 Feb '05

04 Feb '05

This is an announcement for the paper "Minimality, homogeneity and topological 0-1 laws for subspaces of a Banach space" by Valentin Ferenczi. Abstract: If a Banach space is saturated with basic sequences whose linear span embeds into the linear span of any subsequence, then it contains a minimal subspace. It follows that any Banach space is either ergodic or contains a minimal subspace. If $X$ is a Banach space with a Schauder basis, the relation $E_0$ is Borel reducible to permutative equivalence between normalized block-sequences of $X$, or $X$ is $c_0$-saturated or $l_p$-saturated for some $1 \leq p <+\infty$. For a Banach space $X$ with an (unconditional) basis, topological 0-1 law type dichotomies are stated for block-subspaces of $X$ as well as for subspaces of $X$ with a successive FDD on its basis. A uniformity principle for properties of block-sequences, results about block-homogeneity, and a possible method to construct a Banach space with an unconditional basis, which has a complemented subspace without an unconditional basis, are deduced. Archive classification: Functional Analysis; Combinatorics Mathematics Subject Classification: 46B03; 46B15 The source file(s), ferenczitopolaw.tex: 93769 bytes, is(are) stored in gzipped form as 0502054.gz with size 26kb. The corresponding postcript file has gzipped size 99kb. Submitted from: ferenczi(a)ccr.jussieu.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0502054 or http://arXiv.org/abs/math.FA/0502054 or by email in unzipped form by transmitting an empty message with subject line uget 0502054 or in gzipped form by using subject line get 0502054 to: math(a)arXiv.org.

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Abstract of a paper by Harald Hanche-Olsen
by Dale Alspach 04 Feb '05

04 Feb '05

This is an announcement for the paper "On the uniform convexity of L^p" by Harald Hanche-Olsen. Abstract: We present a short, direct proof of the uniform convexity of L^p spaces for 1<p<\infty. Archive classification: Functional Analysis Mathematics Subject Classification: 46E30 The source file(s), uc2.tex: 7706 bytes, is(are) stored in gzipped form as 0502021.gz with size 3kb. The corresponding postcript file has gzipped size 26kb. Submitted from: hanche(a)math.ntnu.no The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0502021 or http://arXiv.org/abs/math.FA/0502021 or by email in unzipped form by transmitting an empty message with subject line uget 0502021 or in gzipped form by using subject line get 0502021 to: math(a)arXiv.org.

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Banach February 2005