Banach

banach@mathdept.okstate.edu

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Abstract of a paper by Boris Rubin
by alspach＠fourier.math.okstate.edu 16 Dec '08

16 Dec '08

This is an announcement for the paper "Comparison of volumes of convex bodies in real, complex, and quaternionic spaces" by Boris Rubin. Abstract: The classical Busemann-Petty problem (1956) asks, whether origin-symmetric convex bodies in $\mathbb {R}^n$ with smaller hyperplane central sections necessarily have smaller volumes. It is known, that the answer is affirmative if $n\le 4$ and negative if $n>4$. The same question can be asked when volumes of hyperplane sections are replaced by more general comparison functions. We give unified exposition of this circle of problems in real, complex, and quaternionic $n$-dimensional spaces. All cases are treated simultaneously. In particular, we show that the Busemann-Petty problem in the quaternionic $n$-dimensional space has an affirmative answer if and only if $n =2$. The method relies on the properties of cosine transforms on the unit sphere. Possible generalizations for spaces over Clifford algebras are discussed. Archive classification: math.FA Mathematics Subject Classification: 44A12; 52A38 Remarks: 38 pages The source file(s), quaternion3.tex: 107627 bytes, is(are) stored in gzipped form as 0812.1300.gz with size 35kb. The corresponding postcript file has gzipped size 182kb. Submitted from: borisr(a)math.lsu.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0812.1300 or http://arXiv.org/abs/0812.1300 or by email in unzipped form by transmitting an empty message with subject line uget 0812.1300 or in gzipped form by using subject line get 0812.1300 to: math(a)arXiv.org.

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Abstract of a paper by R. J. Smith and S. Troyanski
by alspach＠fourier.math.okstate.edu 04 Dec '08

04 Dec '08

This is an announcement for the paper "Unconditional bases and strictly convex dual renormings" by R. J. Smith and S. Troyanski. Abstract: We present equivalent conditions for a space $X$ with an unconditional basis to admit an equivalent norm with a strictly convex dual norm. Archive classification: math.FA Mathematics Subject Classification: 46B03; 46B26; 46B15 The source file(s), unc_basis_dual_sc.tex: 45412 bytes, is(are) stored in gzipped form as 0811.4685.gz with size 14kb. The corresponding postcript file has gzipped size 101kb. Submitted from: smith(a)math.cas.cz The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0811.4685 or http://arXiv.org/abs/0811.4685 or by email in unzipped form by transmitting an empty message with subject line uget 0811.4685 or in gzipped form by using subject line get 0811.4685 to: math(a)arXiv.org.

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Abstract of a paper by Matthew Daws, Hung Le Pham, and Stuart White
by alspach＠fourier.math.okstate.edu 26 Nov '08

26 Nov '08

This is an announcement for the paper "Preduals of semigroup algebras" by Matthew Daws, Hung Le Pham, and Stuart White. Abstract: For a locally compact group $G$, the measure convolution algebra $M(G)$ carries a natural coproduct. In previous work, we showed that the canonical predual $C_0(G)$ of $M(G)$ is the unique predual which makes both the product and the coproduct on $M(G)$ weak$^*$-continuous. Given a discrete semigroup $S$, the convolution algebra $\ell^1(S)$ also carries a coproduct. In this paper we examine preduals for $\ell^1(S)$ making both the product and the coproduct weak$^*$-continuous. Under certain conditions on $S$, we show that $\ell^1(S)$ has a unique such predual. Such $S$ include the free semigroup on finitely many generators. In general, however, this need not be the case even for quite simple semigroups and we construct uncountably many such preduals on $\ell^1(S)$ when $S$ is either $\mathbb Z_+\times\mathbb Z$ or $(\mathbb N,\cdot)$. Archive classification: math.FA Mathematics Subject Classification: 43A20; 22A20 Remarks: 17 pages, LaTeX The source file(s), semigroups.tex: 50737 bytes, is(are) stored in gzipped form as 0811.3987.gz with size 15kb. The corresponding postcript file has gzipped size 114kb. Submitted from: matt.daws(a)cantab.net The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0811.3987 or http://arXiv.org/abs/0811.3987 or by email in unzipped form by transmitting an empty message with subject line uget 0811.3987 or in gzipped form by using subject line get 0811.3987 to: math(a)arXiv.org.

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Abstract of a paper by Geraldo Botelho, Daniel Pellegrino and Pilar Rueda
by alspach＠fourier.math.okstate.edu 26 Nov '08

26 Nov '08

This is an announcement for the paper "A unified Pietsch domination theorem" by Geraldo Botelho, Daniel Pellegrino and Pilar Rueda. Abstract: In this paper we prove an abstract version of Pietsch's domination theorem which unify a number of known Pietsch-type domination theorems for classes of mappings that generalize the ideal of absolutely p-summing linear operators. A final result shows that Pietsch-type dominations are totally free from algebraic conditions, such as linearity, multilinearity, etc. Archive classification: math.FA Remarks: 10 pages The source file(s), abstract-PDT-20nov.tex: 32852 bytes, is(are) stored in gzipped form as 0811.3518.gz with size 9kb. The corresponding postcript file has gzipped size 81kb. Submitted from: dmpellegrino(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0811.3518 or http://arXiv.org/abs/0811.3518 or by email in unzipped form by transmitting an empty message with subject line uget 0811.3518 or in gzipped form by using subject line get 0811.3518 to: math(a)arXiv.org.

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Abstract of a paper by Tuomas Hytonen and Lutz Weis
by alspach＠fourier.math.okstate.edu 26 Nov '08

26 Nov '08

This is an announcement for the paper "The Banach space-valued BMO, Carleson's condition, and paraproducts" by Tuomas Hytonen and Lutz Weis. Abstract: We define a scale of L^q Carleson norms, all of which characterize the membership of a function in BMO. The phenomenon is analogous to the John-Nirenberg inequality, but on the level of Carleson measures. The classical Carleson condition corresponds to the L^2 case in our theory. The result is applied to give a new proof for the L^p-boundedness of paraproducts with a BMO symbol. A novel feature of the argument is that all p are covered at once in a completely interpolation-free manner. This is achieved by using the L^1 Carleson norm, and indicates the usefulness of this notion. Our approach is chosen so that all these results extend in a natural way to the case of X-valued functions, where X is a Banach space with the UMD property. Archive classification: math.FA Mathematics Subject Classification: 42B35; 42B20; 42B25; 46E40 Remarks: 14 pages, submitted The source file(s), carleson.tex: 56068 bytes, is(are) stored in gzipped form as 0811.3333.gz with size 16kb. The corresponding postcript file has gzipped size 106kb. Submitted from: tuomas.hytonen(a)helsinki.fi The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0811.3333 or http://arXiv.org/abs/0811.3333 or by email in unzipped form by transmitting an empty message with subject line uget 0811.3333 or in gzipped form by using subject line get 0811.3333 to: math(a)arXiv.org.

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Abstract of a paper by W. T. Gowers
by alspach＠fourier.math.okstate.edu 26 Nov '08

26 Nov '08

This is an announcement for the paper "Decompositions, approximate structure, transference, and the Hahn-Banach theorem" by W. T. Gowers. Abstract: This paper is partly a survey of certain kinds of results and proofs in additive combinatorics, and partly a discussion of how useful the finite-dimensional Hahn-Banach theorem can be. The most interesting single result is probably a simpler proof of a key step in the proof of the Green-Tao theorem, but several other applications of the method are given. A similarly simplified proof of the Green-Tao transference principle was obtained independently (and expressed in a rather different language) by Reingold, Trevisan, Tulsiani and Vadhan. Archive classification: math.CO math.FA Mathematics Subject Classification: 05D99 Remarks: 48 pages The source file(s), newtransfer6.tex: 157325 bytes, is(are) stored in gzipped form as 0811.3103.gz with size 46kb. The corresponding postcript file has gzipped size 191kb. Submitted from: wtg10(a)dpmms.cam.ac.uk The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0811.3103 or http://arXiv.org/abs/0811.3103 or by email in unzipped form by transmitting an empty message with subject line uget 0811.3103 or in gzipped form by using subject line get 0811.3103 to: math(a)arXiv.org.

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Abstract of a paper by M.I. Ostrovskii
by Dale Alspach 19 Nov '08

19 Nov '08

This is an announcement for the paper "Sufficient enlargements of minimal volume for finite dimensional normed linear spaces" by M.I. Ostrovskii. Abstract: Let $B_Y$ denote the unit ball of a normed linear space $Y$. A symmetric, bounded, closed, convex set $A$ in a finite dimensional normed linear space $X$ is called a {\it sufficient enlargement} for $X$ if, for an arbitrary isometric embedding of $X$ into a Banach space $Y$, there exists a linear projection $P:Y\to X$ such that $P(B_Y)\subset A$. The main results of the paper: {\bf (1)} Each minimal-volume sufficient enlargement is linearly equivalent to a zonotope spanned by multiples of columns of a totally unimodular matrix. {\bf (2)} If a finite dimensional normed linear space has a minimal-volume sufficient enlargement which is not a parallelepiped, then it contains a two-dimensional subspace whose unit ball is linearly equivalent to a regular hexagon. Archive classification: math.FA Mathematics Subject Classification: 46B07, 52A21 Citation: J. Funct. Anal. 255 (2008), no. 3, 589-619 The source file(s), ost.tex: 97543 bytes, is(are) stored in gzipped form as 0811.1701.gz with size 28kb. The corresponding postcript file has gzipped size 173kb. Submitted from: ostrovsm(a)stjohns.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0811.1701 or http://arXiv.org/abs/0811.1701 or by email in unzipped form by transmitting an empty message with subject line uget 0811.1701 or in gzipped form by using subject line get 0811.1701 to: math(a)arXiv.org.

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Abstract of a paper by Oscar Blasco and Jan van Neerven
by alspach＠fourier.math.okstate.edu 19 Nov '08

19 Nov '08

This is an announcement for the paper "Spaces of operator-valued functions measurable with respect to the strong operator topology" by Oscar Blasco and Jan van Neerven. Abstract: Let $X$ and $Y$ be Banach spaces and $(\Omega,\Sigma,\mu)$ a finite measure space. In this note we introduce the space $L^p[\mu;L(X,Y)]$ consisting of all (equivalence classes of) functions $\Phi:\Omega \mapsto L(X,Y)$ such that $\omega \mapsto \Phi(\omega)x$ is strongly $\mu$-measurable for all $x\in X$ and $\omega \mapsto \Phi(\omega)f(\omega)$ belongs to $L^1(\mu;Y)$ for all $f\in L^{p'}(\mu;X)$, $1/p+1/p'=1$. We show that functions in $L^p[\mu;\L(X,Y)]$ define operator-valued measures with bounded $p$-variation and use these spaces to obtain an isometric characterization of the space of all $L(X,Y)$-valued multipliers acting boundedly from $L^p(\mu;X)$ into $L^q(\mu;Y)$, $1\le q< p<\infty$. Archive classification: math.FA Mathematics Subject Classification: 28B05, 46G10 Remarks: 12 pages The source file(s), Blasco_vanNeerven/BlascoVanNeerven.tex: 40452 bytes Blasco_vanNeerven/newsymbol.sty: 440 bytes Blasco_vanNeerven/srcltx.sty: 6955 bytes, is(are) stored in gzipped form as 0811.2284.tar.gz with size 14kb. The corresponding postcript file has gzipped size 97kb. Submitted from: J.M.A.M.vanNeerven(a)tudelft.nl The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0811.2284 or http://arXiv.org/abs/0811.2284 or by email in unzipped form by transmitting an empty message with subject line uget 0811.2284 or in gzipped form by using subject line get 0811.2284 to: math(a)arXiv.org.

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Abstract of a paper by M.I. Ostrovskii
by alspach＠fourier.math.okstate.edu 19 Nov '08

19 Nov '08

This is an announcement for the paper "Compositions of projections in Banach spaces and relations between approximation properties" by M.I. Ostrovskii. Abstract: A necessary and sufficient condition for existence of a Banach space with a finite dimensional decomposition but without the $\pi$-property in terms of norms of compositions of projections is found. Archive classification: math.FA Mathematics Subject Classification: 46B07 Citation: Rocky Mountain Journal of Mathematics, 38 (2008), no. 4, 1253-1262 The source file(s), ostr.tex: 21966 bytes, is(are) stored in gzipped form as 0811.1763.gz with size 7kb. The corresponding postcript file has gzipped size 79kb. Submitted from: ostrovsm(a)stjohns.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0811.1763 or http://arXiv.org/abs/0811.1763 or by email in unzipped form by transmitting an empty message with subject line uget 0811.1763 or in gzipped form by using subject line get 0811.1763 to: math(a)arXiv.org.

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Abstract of a paper by Marisa Zymonopoulou
by alspach＠fourier.math.okstate.edu 19 Nov '08

19 Nov '08

This is an announcement for the paper "A note on the Busemann-Petty problem for bodies of certain invariance" by Marisa Zymonopoulou. Abstract: The Busemann-Petty problem asks whether origin symmetric convex bodies in $\R^n$ with smaller hyperplane sections necessarily have smaller volume. The answer is affirmative if $n\leq 3$ and negative if $n\geq 4.$ We consider a class of convex bodies that have a certain invariance property with respect to their ordered k-tuples of coordinates in $\R^{kn}$ and prove the corresponding problem. Archive classification: math.FA The source file(s), kn.tex: 32692 bytes, is(are) stored in gzipped form as 0811.1593.gz with size 10kb. The corresponding postcript file has gzipped size 82kb. Submitted from: marisa(a)cwru.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0811.1593 or http://arXiv.org/abs/0811.1593 or by email in unzipped form by transmitting an empty message with subject line uget 0811.1593 or in gzipped form by using subject line get 0811.1593 to: math(a)arXiv.org.

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