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3  Extensions of Positive Definite Functions on Amenable Groups Bakonyi, M.; Timotin, D.
Let $S$ be a subset of an amenable group $G$ such that $e\in S$ and
$S^{1}=S$. The main result of this paper states that if the Cayley
graph of $G$ with respect to $S$ has a certain combinatorial property,
then every positive definite operatorvalued function on $S$ can be
extended to a positive definite function on $G$. Several known
extension results are obtained as corollaries. New applications are
also presented.


12  Homotopy and the KestelmanBorweinDitor Theorem Bingham, N. H.; Ostaszewski, A. J.
The KestelmanBorweinDitor Theorem, on embedding a null sequence by
translation in (measure/category) ``large'' sets has two generalizations.
Miller replaces the translated sequence by a ``sequence homotopic
to the identity''. The authors, in a previous paper, replace points by functions:
a uniform functional null sequence replaces the null sequence, and
translation receives a functional form. We give a unified approach to
results of this kind. In particular, we show that (i) Miller's homotopy
version follows from the functional version, and (ii) the pointwise instance
of the functional version follows from Miller's homotopy version.


21  Generalized Dsymmetric Operators II Bouali, S.; Echchad, M.
Let $H$ be a separable,
infinitedimensional, complex Hilbert space and let $A, B\in{\mathcal L
}(H)$, where ${\mathcal L}(H)$ is the algebra of all bounded linear
operators on $H$. Let $\delta_{AB}\colon {\mathcal L}(H)\rightarrow {\mathcal
L}(H)$ denote the generalized derivation $\delta_{AB}(X)=AXXB$.
This note will initiate a study on the class of pairs $(A,B)$ such
that $\overline{{\mathcal R}(\delta_{AB})}= \overline{{\mathcal
R}(\delta_{A^{\ast}B^{\ast}})}$.


28  Generalized Solution of the Photon Transport Problem Chang, YuHsien; Hong, ChengHong
The purpose of this paper is to show the existence of a
generalized solution of the photon transport problem. By means of the theory of
equicontinuous $C_{0}$semigroup on a sequentially complete locally convex
topological vector space we show that the perturbed abstract Cauchy problem
has a unique solution when the perturbation operator and the forcing term
function satisfy certain conditions. A consequence of the abstract result is
that it can be directly applied to obtain a generalized solution of the photon
transport problem.


39  Elements in a Numerical Semigroup with Factorizations of the Same Length Chapman, S. T.; GarcíaSánchez, P. A.; Llena, D.; Marshall, J.
Questions concerning the lengths of factorizations into irreducible
elements in numerical monoids
have gained much attention in the recent literature. In this note,
we show that a numerical monoid has an element with two different
irreducible factorizations of the same length if and only if its
embedding dimension is greater than
two. We find formulas in embedding dimension three for the smallest
element with two different irreducible factorizations of the same
length and the largest element whose different irreducible
factorizations all have distinct lengths. We show that these
formulas do not naturally extend to higher embedding
dimensions.


44  StarShapedness and $K$Orbits in Complex Semisimple Lie Algebras Cheung, WaiShun; Tam, TinYau
Given a complex semisimple Lie algebra
$\mathfrak{g}=\mathfrak{k}+i\mathfrak{k}$ ($\mathfrak{k}$ is a compact
real form of $\mathfrak{g}$), let $\pi\colon\mathfrak{g}\to
\mathfrak{h}$ be the orthogonal projection (with respect to the
Killing form) onto the Cartan subalgebra
$\mathfrak{h}:=\mathfrak{t}+i\mathfrak{t}$, where $\mathfrak{t}$ is a
maximal abelian subalgebra of $\mathfrak{k}$. Given $x\in
\mathfrak{g}$, we consider $\pi(\mathop{\textrm{Ad}}(K) x)$, where $K$ is
the analytic subgroup $G$ corresponding to $\mathfrak{k}$, and show
that it is starshaped. The result extends a result of Tsing. We also
consider the generalized numerical range $f(\mathop{\textrm{Ad}}(K)x)$,
where $f$ is a linear functional on $\mathfrak{g}$. We establish the
starshapedness of $f(\mathop{\textrm{Ad}}(K)x)$ for simple Lie algebras
of type $B$.


56  Characteristic Varieties for a Class of Line Arrangements Dinh, Thi Anh Thu
Let $\mathcal{A}$ be a line arrangement in the complex projective plane
$\mathbb{P}^2$, having the points of multiplicity $\geq 3$ situated on two
lines in $\mathcal{A}$, say $H_0$ and $H_{\infty}$. Then we show that the
nonlocal irreducible components of the first resonance variety
$\mathcal{R}_1(\mathcal{A})$ are 2dimensional and correspond to parallelograms $\mathcal{P}$ in
$\mathbb{C}^2=\mathbb{P}^2 \setminus H_{\infty}$ whose sides are in $\mathcal{A}$ and for
which $H_0$ is a diagonal.


68  Nonsplitting in Kirchberg's Idealrelated $KK$Theory Eilers, Søren; Restorff, Gunnar; Ruiz, Efren
A. Bonkat obtained a universal coefficient theorem in the setting of Kirchberg's
idealrelated $KK$theory in the fundamental case of a
$C^*$algebra with one
specified ideal. The universal coefficient sequence was shown to split, unnaturally, under certain
conditions. Employing certain $K$theoretical information derivable
from the given operator algebras using a method introduced here, we shall
demonstrate that Bonkat's UCT does not split in general. Related
methods lead to information on the complexity of the $K$theory which
must be used to
classify $*$isomorphisms for purely infinite $C^*$algebras with
one nontrivial ideal.


82  Lefschetz Numbers for $C^*$Algebras Emerson, Heath
Using Poincar\'e duality, we obtain a formula of Lefschetz type
that computes the Lefschetz number of an endomorphism of a separable
nuclear $C^*$algebra satisfying Poincar\'e duality and the Kunneth
theorem. (The Lefschetz number of an endomorphism is the graded trace
of the induced map on $\textrm{K}$theory tensored with $\mathbb{C}$, as in the
classical case.) We then examine endomorphisms of CuntzKrieger
algebras $O_A$. An endomorphism has an invariant, which is a
permutation of an infinite set, and the contracting and expanding
behavior of this permutation describes the Lefschetz number of the
endomorphism. Using this description, we derive a closed polynomial
formula for the Lefschetz number depending on the matrix $A$ and the
presentation of the endomorphism.


100  On the Generalized Marcinkiewicz Integral Operators with Rough Kernels Fan, Dashan; Wu, Huoxiong
A class of generalized Marcinkiewicz
integral operators is introduced, and, under rather weak conditions
on the integral kernels, the boundedness of such operators on $L^p$
and TriebelLizorkin spaces is established.


113  On the Norm of the BeurlingAhlfors Operator in Several Dimensions Hytönen, Tuomas P.
The generalized BeurlingAhlfors operator $S$ on
$L^p(\mathbb{R}^n;\Lambda)$, where $\Lambda:=\Lambda(\mathbb{R}^n)$ is the
exterior algebra with its natural Hilbert space norm, satisfies the
estimate
$$\S\_{\mathcal{L}(L^p(\mathbb{R}^n;\Lambda))}\leq(n/2+1)(p^*1),\quad
p^*:=\max\{p,p'\}$$
This improves on earlier results in all dimensions $n\geq 3$. The
proof is based on the heat extension and relies at the bottom on
Burkholder's sharp inequality for martingale transforms.


126  Fundamental Solutions of Kohn SubLaplacians on Anisotropic Heisenberg Groups and Htype Groups Jin, Yongyang; Zhang, Genkai
We prove that the fundamental solutions
of Kohn subLaplacians $\Delta + i\alpha \partial_t$
on the anisotropic Heisenberg groups are tempered distributions and have
meromorphic continuation in $\alpha$ with simple poles. We compute the
residues and find the partial fundamental solutions
at the poles. We also find formulas for the
fundamental solutions for some matrixvalued
Kohn type subLaplacians
on Htype groups.


141  Linear Maps on $C^*$Algebras Preserving the Set of Operators that are Invertible in $\mathcal{A}/\mathcal{I}$ Kim, Sang Og; Park, Choonkil
For $C^*$algebras $\mathcal{A}$ of real rank zero, we describe
linear maps $\phi$ on $\mathcal{A}$ that are surjective up to ideals
$\mathcal{I}$, and $\pi(A)$ is invertible in $\mathcal{A}/\mathcal{I}$ if and only if
$\pi(\phi(A))$ is invertible in $\mathcal{A}/\mathcal{I}$, where $A\in\mathcal{A}$ and
$\pi:\mathcal{A}\to\mathcal{A}/\mathcal{I}$ is the quotient map. We also consider similar
linear maps preserving zero products on the Calkin algebra.


147  Generalized Quandle Polynomials Nelson, Sam
We define a family of generalizations of the twovariable quandle polynomial.
These polynomial invariants generalize in a natural way to eightvariable
polynomial invariants of finite biquandles. We use these polynomials to define
a family of link invariants that further generalize the quandle counting
invariant.


159  Hardy Inequalities on the Real Line Sababheh, Mohammad
We prove that some inequalities, which are considered to be
generalizations of Hardy's inequality on the circle,
can be modified and proved to be true for functions integrable on the real line.
In fact we would like to show that some constructions that were
used to prove the Littlewood conjecture can be used similarly to
produce real Hardytype inequalities.
This discussion will lead to many questions concerning the
relationship between Hardytype inequalities on the circle and
those on the real line.


172  Measures with Fourier Transforms in $L^2$ of a Halfspace Shayya, Bassam
We prove that if the Fourier transform of a compactly supported
measure is in $L^2$ of a halfspace, then the measure is
absolutely continuous to Lebesgue measure. We then show how this
result can be used to translate information about the
dimensionality of a measure and the decay of its Fourier
transform into geometric information about its support.


180  Additive Families of Low Borel Classes and Borel Measurable Selectors Spurný, J.; Zelený, M.
An important conjecture in the theory of Borel sets in nonseparable
metric spaces is whether any pointcountable Boreladditive family in
a complete metric space has a $\sigma$discrete refinement. We confirm the conjecture for
pointcountable $\mathbf\Pi_3^0$additive families, thus generalizing results of
R. W. Hansell and the first author. We apply this result to the
existence of Borel measurable selectors for multivalued mappings of
low Borel complexity, thus answering in the affirmative a particular
version of a question of J. Kaniewski and R. Pol.

