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
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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
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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
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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
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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
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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
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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
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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
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This is an announcement for the paper "General Hormander and Mikhlin
conditions for multipliers of Besov spaces" by Rishad Shahmurov.
Abstract: Here a new condition for the geometry of Banach spaces is
introduced and the operator--valued Fourier multiplier theorems in
weighted Besov spaces are obtained. Particularly, connections between
the geometry of Banach spaces and Hormander-Mikhlin conditions are
established. As an application of main results the regularity properties
of degenerate elliptic differential operator equations are investigated.
Archive classification: math.FA math.CA
Mathematics Subject Classification: 34G10, 35J25, 35J70
Remarks: 16
The source file(s), FMTWeightedB.tex: 57462 bytes, is(are) stored in
gzipped form as 0811.1350.gz with size 14kb. The corresponding postcript
file has gzipped size 101kb.
Submitted from: shahmurov(a)hotmail.com
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This is an announcement for the paper "Convexity and smoothness of Banach
spaces with numerical index one" by Vladimir Kadets, Miguel Martin,
Javier Meri, and Rafael Paya .
Abstract: We show that a Banach space with numerical index one
cannot enjoy good convexity or smoothness properties unless it is
one-dimensional. For instance, it has no WLUR points in its unit ball,
its norm is not Frechet smooth and its dual norm is neither smooth nor
strictly convex. Actually, these results also hold if the space has
the (strictly weaker) alternative Daugavet property. We construct a
(non-complete) strictly convex predual of an infinite-dimensional $L_1$
space (which satisfies a property called lushness which implies numerical
index~$1$). On the other hand, we show that a lush real Banach space is
neither strictly convex nor smooth, unless it is one-dimensional. In
particular, if a subspace $X$ of the real space $C[0,1]$ is smooth or
strictly convex, then $C[0,1]/X$ contains a copy of $C[0,1]$. Finally,
we prove that the dual of any lush infinite-dimensional real space
contains a copy of $\ell_1$.
Archive classification: math.FA math.OA
Mathematics Subject Classification: 46B04, 46B20, 47A12
Remarks: Illinois J. Math. (to appear)
The source file(s), Kadets-Martin-Meri-Paya.tex: 61549 bytes, is(are)
stored in gzipped form as 0811.0808.gz with size 19kb. The corresponding
postcript file has gzipped size 120kb.
Submitted from: mmartins(a)ugr.es
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