






« 2002 (v45)  2004 (v47) » 
Page 


3  Condensed Domains Anderson, D. D.; Dumitrescu, Tiberiu
An integral domain $D$ with identity is condensed (resp., strongly
condensed) if for each pair of ideals $I$, $J$ of $D$, $IJ=\{ij; i\in I,
j\in J\}$ (resp., $IJ=iJ$ for some $i\in I$ or $IJ =Ij$ for some
$j\in J$). We show that for a Noetherian domain $D$, $D$ is condensed
if and only if $\Pic(D)=0$ and $D$ is locally condensed, while a local
domain is strongly condensed if and only if it has the twogenerator
property. An integrally closed domain $D$ is strongly condensed if and
only if $D$ is a B\'{e}zout generalized Dedekind domain with at most one
maximal ideal of height greater than one. We give a number of
equivalencies for a local domain with finite integral closure to be
strongly condensed. Finally, we show that for a field extension
$k\subseteq K$, the domain $D=k+XK[[X]]$ is condensed if and only if
$[K:k]\leq 2$ or $[K:k]=3$ and each degreetwo polynomial in $k[X]$
splits over $k$, while $D$ is strongly condensed if and only if $[K:k]
\leq 2$.


14  Generalized Commutativity in Group Algebras Bahturin, Yu. A.; Parmenter, M. M.
We study group algebras $FG$ which can be graded by a finite abelian
group $\Gamma$ such that $FG$ is $\beta$commutative for a
skewsymmetric bicharacter $\beta$ on $\Gamma$ with values in $F^*$.


26  Remarques sur les points rationnels des variétés de Fermat Bernardi, D.; Halberstadt, E.; Kraus, A.
Soit $K$ un corps de nombres de degr\'e sur $\mathbb{Q}$ inf\'erieur
ou \'egal \`a $2$. On se propose dans ce travail de faire quelques
remarques sur la question de l'existence de deux \'el\'ements non nuls
$a$ et $b$ de $K$, et d'un entier $n\geq 4$, tels que l'\'equation
$ax^n + by^n = 1$ poss\`ede au moins trois points distincts non
triviaux. Cette \'etude se ram\`ene \`a la recherche de points
rationnels sur $K$ d'une vari\'et\'e projective dans $\mathbb{P}^5$ de
dimension $3$, ou d'une surface de $\mathbb{P}^3$.


39  Power Residue Criteria for Quadratic Units and the Negative Pell Equation Bülow, Tommy
Let $d>1$ be a squarefree integer. Power residue criteria for the
fundamental unit $\varepsilon_d$ of the real quadratic fields $\QQ
(\sqrt{d})$ modulo a prime $p$ (for certain $d$ and $p$) are proved by
means of class field theory. These results will then be interpreted
as criteria for the solvability of the negative Pell equation $x^2 
dp^2 y^2 = 1$. The most important solvability criterion deals with
all $d$ for which $\QQ (\sqrt{d})$ has an elementary abelian 2class
group and $p\equiv 5\pmod{8}$ or $p\equiv 9\pmod{16}$.


54  Linear Maps Transforming the Unitary Group Cheung, WaiShun; Li, ChiKwong
Let $U(n)$ be the group of $n\times n$ unitary matrices. We show that if
$\phi$ is a linear transformation sending $U(n)$ into $U(m)$, then $m$ is
a multiple of $n$, and $\phi$ has the form
$$
A \mapsto V[(A\otimes I_s)\oplus (A^t \otimes I_{r})]W
$$
for some $V, W \in U(m)$. From this result, one easily deduces the
characterization of linear operators that map $U(n)$ into itself obtained
by Marcus. Further generalization of the main theorem is also discussed.


59  A Note on Noncommutative Interpolation Constantinescu, T.; Johnson, J. L.
In this paper we formulate and solve NevanlinnaPick and
Carath\'eodory type problems for tensor algebras with data given on
the $N$dimensional operator unit ball of a Hilbert space. We develop
an approach based on the displacement structure theory.


71  The Number of Fields Generated by the Square Root of Values of a Given Polynomial Cutter, Pamela; Granville, Andrew; Tucker, Thomas J.
The $abc$conjecture is applied to various questions involving the
number of distinct fields $\mathbb{Q} \bigl( \sqrt{f(n)} \bigr)$, as we vary
over integers $n$.


80  MultiSided Braid Type Subfactors, II Erlijman, Juliana
We show that the multisided inclusion $R^{\otimes l} \subset R$ of
braidtype subfactors of the hyperfinite II$_1$ factor $R$, introduced
in {\it Multisided braid type subfactors} [E3], contains a sequence
of intermediate subfactors: $R^{\otimes l} \subset R^{\otimes l1}
\subset \cdots \subset R^{\otimes 2} \subset R$. That is, every
$t$sided subfactor is an intermediate subfactor for the inclusion
$R^{\otimes l} \subset R$, for $2\leq t\leq l$. Moreover, we also
show that if $t>m$ then $R^{\otimes t} \subset R^{\otimes m}$ is
conjugate to $R^{\otimes tm+1} \subset R$. Thus, if the braid
representation considered is associated to one of the classical Lie
algebras then the asymptotic inclusions for the JonesWenzl subfactors
are intermediate subfactors.


95  Cercles de remplissage for the Riemann Zeta Function Gauthier, P. M.
The celebrated theorem of Picard asserts that each nonconstant entire
function assumes every value infinitely often, with at most one
exception. The Riemann zeta function has this Picard behaviour in a
sequence of discs lying in the critical band and whose diameters tend
to zero. According to the Riemann hypothesis, the value zero would be
this (unique) exceptional value.


98  Crossed Products by Semigroups of Endomorphisms and Groups of Partial Automorphisms Larsen, Nadia S.
We consider a class $(A, S, \alpha)$ of dynamical systems,
where $S$ is an Ore semigroup and $\alpha$ is an action such that
each $\alpha_s$ is injective and extendible ({\it i.e.} it extends to a
nonunital endomorphism of the multiplier algebra), and has range an
ideal of $A$. We show that there is a partial action on the fixedpoint
algebra under the canonical coaction of the enveloping group $G$ of $S$
constructed in \cite[Proposition~6.1]{LR2}. It turns out that the full
crossed product by this coaction is isomorphic to $A\rtimes_\alpha S$.
If the coaction is moreover normal, then the isomorphism can be extended
to include the reduced crossed product. We look then at invariant ideals
and finally, at examples of systems where our results apply.


113  Properties of the $\mathcal{M}$Harmonic Conjugate Operator Lee, Jaesung; Rim, Kyung Soo
We define the $\mathcal{M}$harmonic conjugate operator $K$ and
prove that it is bounded on the nonisotropic Lipschitz space and on
$\BMO$. Then we show $K$ maps Dini functions into the space of
continuous functions on the unit sphere. We also prove the
boundedness and compactness properties of $\mathcal{M}$harmonic
conjugate operator with $L^p$ symbol.


122  On Certain Finitely Generated Subgroups of Groups Which Split Moon, Myoungho
Define a group $G$ to be in the class $\mathcal{S}$ if for any
finitely generated subgroup $K$ of $G$ having the property that
there is a positive integer $n$ such that $g^n \in K$ for all
$g\in G$, $K$ has finite index in $G$. We show that a free
product with amalgamation $A*_C B$ and an $\HNN$ group $A *_C$ belong
to $\mathcal{S}$, if $C$ is in $\mathcal{S}$ and every subgroup of
$C$ is finitely generated.


130  On Frankel's Theorem Petersen, Peter; Wilhelm, Frederick
In this paper we show that two minimal hypersurfaces in a manifold with
positive Ricci curvature must intersect. This is then generalized to show
that in manifolds with positive Ricci curvature in the integral sense two
minimal hypersurfaces must be close to each other. We also show
what happens if a manifold with nonnegative Ricci curvature admits
two nonintersecting minimal hypersurfaces.


140  An Explicit Cell Decomposition of the Wonderful Compactification of a Semisimple Algebraic Group Renner, Lex E.
We determine an explicit cell decomposition of the wonderful
compactification of a semi\simple algebraic group. To do this we first
identify the $B\times B$orbits using the generalized Bruhat
decomposition of a reductive monoid. From there we show how each cell
is made up from $B\times B$orbits.


149  The Ramification Polygon for Curves over a Finite Field Scherk, John
A Newton polygon is introduced for a ramified point of a Galois
covering of curves over a finite field. It is shown to be determined
by the sequence of higher ramification groups of the point. It gives
a blowing up of the wildly ramified part which separates the branches
of the curve. There is also a connection with local reciprocity.


157  Torsion Points on Certain Families of Elliptic Curves Wieczorek, Małgorzata
Fix an elliptic curve $y^2 = x^3+Ax+B$, satisfying $A,B \in \ZZ$,
$A\geq B > 0$. We prove that the $\QQ$torsion subgroup is one of
$(0)$, $\ZZ/3\ZZ$, $\ZZ/9\ZZ$. Related numerical calculations are
discussed.


161  Answer to a Question of S.~Rolewicz Cabello Sánchez, Félix; Castillo, Jesús M. F.
We exhibit examples of $F$spaces with trivial dual which are
isomorphic to its quotient by a line, thus solving a problem in
Rolewicz's {\it Metric Linear Spaces}.


164  Classification of $\AF$ Flows Dean, Andrew J.
An $\AF$ flow is a oneparameter automorphism group of an $\AF$
$C^*$algebra $A$ such that there exists an increasing sequence of
invariant finite dimensional sub$C^*$algebras whose union is dense in
$A$. In this paper, a classification of $C^*$dynamical systems of
this form up to equivariant isomorphism is presented. Two pictures
of the actions are given, one in terms of a modified Bratteli
diagram/pathspace construction, and one in terms of a modified
$K_0$ functor.


178  Sur les invariants d'Iwasawa des tours cyclotomiques Jaulent, JeanFrançois; Maire, Christian
We carry out the computation of the Iwasawa invariants $\rho^T_S$,
$\mu^T_S$, $\lambda^T_S$ associated to abelian $T$ramified
over the finite steps $K_n$ of the cyclotomic
$\mathbb{Z}_\ell$extension $K_\infty/K$ of a number field of
$\CM$type.


191  Weak Type Estimates of the Maximal Quasiradial BochnerRiesz Operator On Certain Hardy Spaces Kim, YongCheol
Let $\{A_t\}_{t>0}$ be the dilation group in $\mathbb{R}^n$ generated
by the infinitesimal generator $M$ where $A_t=\exp(M\log t)$, and let
$\varrho\in C^{\infty}(\mathbb{R}^n\setminus\{0\})$ be a
$A_t$homogeneous distance function defined on $\mathbb{R}^n$. For
$f\in \mathfrak{S}(\mathbb{R}^n)$, we define the maximal quasiradial
BochnerRiesz operator $\mathfrak{M}^{\delta}_{\varrho}$ of index
$\delta>0$ by
$$
\mathfrak{M}^{\delta}_{\varrho} f(x)=\sup_{t>0}\mathcal{F}^{1}
[(1\varrho/t)_+^{\delta}\hat f ](x).
$$
If $A_t=t I$ and $\{\xi\in \mathbb{R}^n\mid \varrho(\xi)=1\}$ is a
smooth convex hypersurface of finite type, then we prove in an
extremely easy way that $\mathfrak{M}^{\delta}_{\varrho}$ is well
defined on $H^p(\mathbb{R}^n)$ when $\delta=n(1/p1/2)1/2$ and
$0<p<1$; moreover, it is a bounded operator from $H^p(\mathbb{R}^n)$
into $L^{p,\infty}(\mathbb{R}^n)$.
If $A_t=t I$ and $\varrho\in C^{\infty}(\mathbb{R}^n\setminus\{0\})$,
we also prove that $\mathfrak{M}^{\delta}_{\varrho}$ is a bounded
operator from $H^p(\mathbb{R}^n)$ into $L^p(\mathbb{R}^n)$ when
$\delta>n(1/p1/2)1/2$ and $0<p<1$.


204  Rationality and Orbit Closures Levy, Jason
Suppose we are given a finitedimensional vector space $V$ equipped
with an $F$rational action of a linearly algebraic group $G$, with
$F$ a characteristic zero field. We conjecture the following: to each
vector $v\in V(F)$ there corresponds a canonical $G(F)$orbit of
semisimple vectors of $V$. In the case of the adjoint action, this
orbit is the $G(F)$orbit of the semisimple part of $v$, so this
conjecture can be considered a generalization of the Jordan
decomposition. We prove some cases of the conjecture.


216  Linear Maps on Selfadjoint Operators Preserving Invertibility, Positive Definiteness, Numerical Range Li, ChiKwong; Rodman, Leiba; Šemrl, Peter
Let $H$ be a complex Hilbert space, and $\HH$ be the real linear space of
bounded selfadjoint operators on $H$. We study linear maps $\phi\colon \HH
\to \HH$ leaving invariant various properties such as invertibility, positive
definiteness, numerical range, {\it etc}. The maps $\phi$ are not assumed
{\it a priori\/} continuous. It is shown that under an appropriate surjective
or injective assumption $\phi$ has the form $X \mapsto \xi TXT^*$ or $X \mapsto
\xi TX^tT^*$, for a suitable invertible or unitary $T$ and $\xi\in\{1, 1\}$,
where $X^t$ stands for the transpose of $X$ relative to some orthonormal basis.
Examples are given to show that the surjective or injective assumption cannot
be relaxed. The results are extended to complex linear maps on the algebra of
bounded linear operators on $H$. Similar results are proved for the (real)
linear space of (selfadjoint) operators of the form $\alpha I+K$, where $\alpha$
is a scalar and $K$ is compact.


229  Counting the Number of Integral Points in General $n$Dimensional Tetrahedra and Bernoulli Polynomials Lin, KePao; Yau, Stephen S.T.
Recently there has been tremendous interest in counting the number of
integral points in $n$dimen\sional tetrahedra with nonintegral
vertices due to its applications in primality testing and factoring
in number theory and in singularities theory. The purpose of this
note is to formulate a conjecture on sharp upper estimate of the
number of integral points in $n$dimensional tetrahedra with
nonintegral vertices. We show that this conjecture is true for
low dimensional cases as well as in the case of homogeneous
$n$dimensional tetrahedra. We also show that the Bernoulli
polynomials play a role in this counting.


242  Euclidean Sections of Direct Sums of Normed Spaces Litvak, A. E.; Milman, V. D.
We study the dimension of ``random'' Euclidean sections of direct sums of
normed spaces. We compare the obtained results with results from \cite{LMS},
to show that for the direct sums the standard randomness with respect to the
Haar measure on Grassmanian coincides with a much ``weaker'' randomness of
``diagonal'' subspaces (Corollary~\ref{sle} and explanation after). We also
add some relative information on ``phase transition''.


252  BeurlingDahlbergSjögren Type Theorems for Minimally Thin Sets in a Cone Miyamoto, Ikuko; Yanagishita, Minoru; Yoshida, Hidenobu
This paper shows that some characterizations of minimally thin sets
connected with a domain having smooth boundary and a halfspace in
particular also hold for the minimally thin sets at a corner point of
a special domain with corners, {\it i.e.}, the minimally thin set at
$\infty$ of a cone.


265  Reducing Spheres and Klein Bottles after Dehn Fillings Oh, Seungsang
Let $M$ be a compact, connected, orientable, irreducible 3manifold with a
torus boundary. It is known that if two Dehn fillings on $M$ along the
boundary produce a reducible manifold and a manifold containing a Klein
bottle, then the distance between the filling slopes is at most three. This
paper gives a remarkably short proof of this result.


268  Group Cohomology and $L^p$Cohomology of Finitely Generated Groups Puls, Michael J.
Let $G$ be a finitely generated, infinite group, let $p>1$, and let
$L^p(G)$ denote the Banach space $\{ \sum_{x\in G} a_xx \mid \sum_{x\in
G} a_x ^p < \infty \}$. In this paper we will study the first
cohomology group of $G$ with coefficients in $L^p(G)$, and the first
reduced $L^p$cohomology space of $G$. Most of our results will be for a
class of groups that contains all finitely generated, infinite nilpotent
groups.


277  Rigidity of Hamiltonian Actions Rochon, Frédéric
This paper studies the following question: Given an
$\omega'$symplectic action of a Lie group on a manifold $M$ which
coincides, as a smooth action, with a Hamiltonian $\omega$action,
when is this action a Hamiltonian $\omega'$action? Using a result of
MorseBott theory presented in Section~2, we show in Section~3 of this
paper that such an action is in fact a Hamiltonian $\omega'$action,
provided that $M$ is compact and that the Lie group is compact and
connected. This result was first proved by LalondeMcDuffPolterovich
in 1999 as a consequence of a more general theory that made use of
hard geometric analysis. In this paper, we prove it using classical
methods only.


291  A Coincidence Theorem for Holomorphic Maps to $G/P$ Sankaran, Parameswaran
The purpose of this note is to extend to an arbitrary generalized Hopf
and CalabiEckmann manifold the following result of Kalyan Mukherjea:
Let $V_n = \mathbb{S}^{2n+1} \times \mathbb{S}^{2n+1}$ denote a
CalabiEckmann manifold. If $f,g \colon V_n \longrightarrow
\mathbb{P}^n$ are any two holomorphic maps, at least one of them being
nonconstant, then there exists a coincidence: $f(x)=g(x)$ for some
$x\in V_n$. Our proof involves a coincidence theorem for holomorphic
maps to complex projective varieties of the form $G/P$ where $G$ is
complex simple algebraic group and $P\subset G$ is a maximal parabolic
subgroup, where one of the maps is dominant.


299  A Basis of Bachmuth Type in the Commutator Subgroup of a Free Group Tomaszewski, Witold
We show here that the commutator subgroup of a free group of finite
rank poses a basis of Bachmuth's type.


304  Localization of the HasseSchmidt Algebra Traves, William N.
The behaviour of the HasseSchmidt algebra of higher derivations under
localization is studied using Andr\'e cohomology. Elementary
techniques are used to describe the HasseSchmidt derivations on
certain monomial rings in the nonmodular case. The localization
conjecture is then verified for all monomial rings.


310  Second Order Dehn Functions of Asynchronously Automatic Groups Wang, Xiaofeng
Upper bounds of second order Dehn functions of asynchronously
automatic groups are obtained.


321  Discreteness For the Set of Complex Structures On a Real Variety Ballico, E.
Let $X$, $Y$ be reduced and irreducible compact complex spaces and
$S$ the set of all isomorphism classes of reduced and irreducible
compact complex spaces $W$ such that $X\times Y \cong X\times W$.
Here we prove that $S$ is at most countable. We apply this result
to show that for every reduced and irreducible compact complex
space $X$ the set $S(X)$ of all complex reduced compact complex
spaces $W$ with $X\times X^\sigma \cong W\times W^\sigma$ (where
$A^\sigma$ denotes the complex conjugate of any variety $A$) is at
most countable.


323  Characterizing TwoDimensional Maps Whose Jacobians Have Constant Eigenvalues Chamberland, Marc
Recent papers have shown that $C^1$ maps $F\colon \mathbb{R}^2
\rightarrow \mathbb{R}^2$
whose Jacobians have constant eigenvalues can be completely
characterized if either the eigenvalues are equal or $F$ is a
polynomial. Specifically, $F=(u,v)$ must take the form
\begin{gather*}
u = ax + by + \beta \phi(\alpha x + \beta y) + e \\
v = cx + dy  \alpha \phi(\alpha x + \beta y) + f
\end{gather*}
for some constants $a$, $b$, $c$, $d$, $e$, $f$, $\alpha$, $\beta$ and
a $C^1$ function $\phi$ in one variable. If, in addition, the function
$\phi$ is not affine, then
\begin{equation}
\alpha\beta (da) + b\alpha^2  c\beta^2 = 0.
\end{equation}
This paper shows how these theorems cannot be extended by constructing
a realanalytic map whose Jacobian eigenvalues are $\pm 1/2$ and does
not fit the previous form. This example is also used to construct
nonobvious solutions to nonlinear PDEs, including the MongeAmp\`ere
equation.


332  Some Questions about Semisimple Lie Groups Originating in Matrix Theory Đoković, Dragomir Z.; Tam, TinYau
We generalize the wellknown result that a square traceless complex
matrix is unitarily similar to a matrix with zero diagonal to
arbitrary connected semisimple complex Lie groups $G$ and their Lie
algebras $\mathfrak{g}$ under the action of a maximal compact subgroup
$K$ of $G$. We also introduce a natural partial order on
$\mathfrak{g}$: $x\le y$ if $f(K\cdot x) \subseteq f(K\cdot y)$ for
all $f\in \mathfrak{g}^*$, the complex dual of $\mathfrak{g}$. This
partial order is $K$invariant and induces a partial order on the
orbit space $\mathfrak{g}/K$. We prove that, under some restrictions
on $\mathfrak{g}$, the set $f(K\cdot x)$ is starshaped with respect
to the origin.


344  Gauss and Eisenstein Sums of Order Twelve Gurak, S.
Let $q=p^{r}$ with $p$ an odd prime, and $\mathbf{F}_{q}$ denote the finite
field of $q$ elements. Let $\Tr\colon\mathbf{F}_{q} \to\mathbf{F}_{p} $ be
the usual trace map and set $\zeta_{p} =\exp(2\pi i/p)$. For any positive
integer $e$, define the (modified) Gauss sum $g_{r}(e)$ by
$$
g_{r}(e) =\sum_{x\in \mathbf{F}_{q}}\zeta_{p}^{\Tr x^{e}}
$$
Recently, Evans gave an elegant determination of $g_{1}(12)$ in terms of
$g_{1}(3)$, $g_{1}(4)$ and $g_{1}(6)$ which resolved a sign ambiguity
present in a previous evaluation. Here I generalize Evans' result to give
a complete determination of the sum $g_{r}(12)$.


356  Branched Covers of Tangles in Threeballs Ishiwata, Makiko; Przytycki, Józef H.; Yasuhara, Akira
We give an algorithm for a surgery description of a $p$fold cyclic branched
cover of $B^3$ branched along a tangle. We generalize constructions of
Montesinos and AkbulutKirby.


365  Homogeneity of the Pure State Space of a Separable $C^*$Algebra Kishimoto, Akitaka; Ozawa, Narutaka; Sakai, Shôichirô
We prove that the pure state space is homogeneous under the action of
the automorphism group (or the subgroup of asymptotically inner
automorphisms) for all the separable simple $C^*$algebras. The
first result of this kind was shown by Powers for the UHF algbras
some 30 years ago.


373  Potential Theory of the FarthestPoint Distance Function Laugesen, Richard S.; Pritsker, Igor E.
We study the farthestpoint distance function, which measures the
distance from $z \in \mathbb{C}$ to the farthest point or points of
a given compact set $E$ in the plane.


388  Tracially Quasidiagonal Extensions Lin, Huaxin
It is known that a unital simple $C^*$algebra $A$ with tracial
topological rank zero has real rank zero. We show in this note that,
in general, there are unital $C^*$algebras with tracial topological
rank zero that have real rank other than zero.


400  Approximating Positive Polynomials Using Sums of Squares Marshall, M.
The paper considers the relationship between positive polynomials,
sums of squares and the multidimensional moment problem in the
general context of basic closed semialgebraic sets in real $n$space.
The emphasis is on the noncompact case and on quadratic module
representations as opposed to quadratic preordering presentations.
The paper clarifies the relationship between known results on the
algebraic side and on the functionalanalytic side and extends these
results in a variety of ways.


419  On NonStrongly Free Automorphisms of Subfactors of Type III$_0$ Masuda, Toshihiko
We determine when an automorphism of a subfactor of type III$_0$
with finite index is nonstrongly free in the sense of C.~Winsl\o w
in terms of the modular endomorphisms introduced by M.~Izumi.


429  The Grothendieck Trace and the de Rham Integral Sastry, Pramathanath; Tong, Yue Lin L.
On a smooth $n$dimensional complete variety $X$ over ${\mathbb C}$ we
show that the trace map ${\tilde\theta}_X \colon\break
H^n (X,\Omega_X^n)
\to {\mathbb C}$ arising from Lipman's version of Grothendieck duality
in \cite{ast117} agrees with
$$
(2\pi i)^{n} (1)^{n(n1)/2} \int_X \colon H^{2n}_{DR} (X,{\mathbb
C}) \to {\mathbb C}
$$
under the Dolbeault isomorphism.


441  An Inductive Limit Model for the $K$Theory of the GeneratorInterchanging Antiautomorphism of an Irrational Rotation Algebra Stacey, P. J.
Let $A_\theta$ be the universal $C^*$algebra generated by two
unitaries $U$, $V$ satisfying $VU=e^{2\pi i\theta} UV$ and let $\Phi$
be the antiautomorphism of $A_\theta$ interchanging $U$ and $V$. The
$K$theory of $R_\theta=\{a\in A_\theta:\Phi(a)=a^*\}$ is computed. When
$\theta$ is irrational, an inductive limit of algebras of the form
$M_q(C(\mathbb{T})) \oplus M_{q'} (\mathbb{R}) \oplus M_q(\mathbb{R})$
is constructed which has complexification $A_\theta$ and the same
$K$theory as $R_\theta$.


457  Strongly Perforated $K_{0}$Groups of Simple $C^{*}$Algebras Toms, Andrew
In the sequel we construct simple, unital, separable, stable, amenable
$C^{*}$algebras for which the ordered $K_{0}$group is strongly
perforated and group isomorphic to $Z$. The particular order structures
to be constructed will be described in detail below, and all
known results of this type will be generalised.


473  A Multiplicative Analogue of Schur's Tauberian Theorem Yeats, Karen
A theorem concerning the asymptotic behaviour of partial sums of the
coefficients of products of Dirichlet series is proved using properties of
regularly varying functions. This theorem is a multiplicative analogue of
Schur's Tauberian theorem for power series.


481  On the Composition of Differentiable Functions Bachir, M.; Lancien, G.
We prove that a Banach space $X$ has the Schur property if and only if every
$X$valued weakly differentiable function is Fr\'echet differentiable. We
give a general result on the Fr\'echet differentiability of $f\circ T$, where
$f$ is a Lipschitz function and $T$ is a compact linear operator. Finally
we study, using in particular a smooth variational principle, the
differentiability of the semi norm $\Vert \ \Vert_{\lip}$ on various spaces
of Lipschitz functions.


495  Canonical Vector Heights on Algebraic K3 Surfaces with Picard Number Two Baragar, Arthur
Let $V$ be an algebraic K3 surface defined over a number field $K$.
Suppose $V$ has Picard number two and an infinite group of
automorphisms $\mathcal{A} = \Aut(V/K)$. In this paper, we
introduce the notion of a vector height $\mathbf{h} \colon V \to
\Pic(V) \otimes \mathbb{R}$ and show the existence of a canonical
vector height $\widehat{\mathbf{h}}$ with the following properties:
\begin{gather*}
\widehat{\mathbf{h}} (\sigma P) = \sigma_* \widehat{\mathbf{h}} (P) \\
h_D (P) = \widehat{\mathbf{h}} (P) \cdot D + O(1),
\end{gather*}
where $\sigma \in \mathcal{A}$, $\sigma_*$ is the pushforward of
$\sigma$ (the pullback of $\sigma^{1}$), and $h_D$ is a Weil
height associated to the divisor $D$. The bounded function implied
by the $O(1)$ does not depend on $P$. This allows us to attack
some arithmetic problems. For example, we show that the number of
rational points with bounded logarithmic height in an
$\mathcal{A}$orbit satisfies
$$
N_{\mathcal{A}(P)} (t,D) = \# \{Q \in \mathcal{A}(P) : h_D (Q)<t\}
= \frac{\mu(P)}{s\log \omega} \log t + O \Bigl( \log \bigl(
\widehat{\mathbf{h}} (P) \cdot D+2 \bigr) \Bigr).
$$
Here, $\mu(P)$ is a nonnegative integer, $s$ is a positive integer,
and $\omega$ is a real quadratic fundamental unit.


509  Symmetries of Kirchberg Algebras Benson, David J.; Kumjian, Alex; Phillips, N. Christopher
Let $G_0$ and $G_1$ be countable abelian groups. Let $\gamma_i$ be an
automorphism of $G_i$ of order two. Then there exists a unital
Kirchberg algebra $A$ satisfying the Universal Coefficient Theorem and
with $[1_A] = 0$ in $K_0 (A)$, and an automorphism $\alpha \in
\Aut(A)$ of order two, such that $K_0 (A) \cong G_0$, such that $K_1
(A) \cong G_1$, and such that $\alpha_* \colon K_i (A) \to K_i (A)$ is
$\gamma_i$. As a consequence, we prove that every
$\mathbb{Z}_2$graded countable module over the representation ring $R
(\mathbb{Z}_2)$ of $\mathbb{Z}_2$ is isomorphic to the equivariant
$K$theory $K^{\mathbb{Z}_2} (A)$ for some action of $\mathbb{Z}_2$ on
a unital Kirchberg algebra~$A$.


529  Representations of the Twisted HeisenbergVirasoro Algebra at Level Zero Billig, Yuly
We describe the structure of the irreducible highest weight modules
for the twisted HeisenbergVirasoro Lie algebra at level zero. We
prove that either a Verma module is irreducible or its maximal
submodule is cyclic.


538  Subdifferentials Whose Graphs Are Not Norm$\times$Weak* Closed Borwein, Jonathan; Fitzpatrick, Simon; Girgensohn, Roland
In this note we give examples of convex functions whose
subdifferentials have unpleasant properties. Particularly, we
exhibit a proper lower semicontinuous convex function on a
separable Hilbert space such that the graph of its subdifferential
is not closed in the product of the norm and bounded weak
topologies. We also exhibit a set whose sequential normal cone is
not norm closed.


546  $L$Series of Certain Elliptic Surfaces Long, Ling
In this paper, we study the modularity of certain elliptic surfaces
by determining their $L$series through their monodromy groups.


559  On Density Conditions for Interpolation in the Ball Marco, Nicolas; Massaneda, Xavier
In this paper we study interpolating sequences for two related spaces of
holomorphic functions in the unit ball of $\C^n$, $n>1$. We first give density
conditions for a sequence to be interpolating for the class $A^{\infty}$ of
holomorphic functions with polynomial growth. The sufficient condition is
formally identical to the characterizing condition in dimension $1$, whereas the
necessary one goes along the lines of the results given by Li and Taylor for
some spaces of entire functions. In the second part of the paper we show that a
density condition, which for $n=1$ coincides with the characterizing condition
given by Seip, is sufficient for interpolation in the (weighted) Bergman space.


575  Optimization of Polynomial Functions Marshall, M.
This paper develops a refinement of Lasserre's algorithm for
optimizing a polynomial on a basic closed semialgebraic set via
semidefinite programming and addresses an open question concerning the
duality gap. It is shown that, under certain natural stability
assumptions, the problem of optimization on a basic closed set reduces
to the compact case.


588  Weakly Stable Relations and Inductive Limits of $C^\ast$algebras Monteiro, Martha Salerno
We show that if $\mathcal{A}$ is a class of $C^\ast$algebras for which
the set of formal relations $\mathcal{R}$ is weakly stable, then $\mathcal{R}$
is weakly stable for the class $\mathcal{B}$ that contains $\mathcal{A}$ and
all the inductive limits that can be constructed with the $C^\ast$algebras in
$\mathcal{A}$.


597  Cartan Subalgebras of $\mathfrak{gl}_\infty$ Neeb, KarlHermann; Penkov, Ivan
Let $V$ be a vector space over a field $\mathbb{K}$ of characteristic
zero and $V_*$ be a space of linear functionals on $V$ which separate
the points of $V$. We consider $V\otimes V_*$ as a Lie algebra of
finite rank operators on $V$, and set $\mathfrak{gl} (V,V_*) :=
V\otimes V_*$. We define a Cartan subalgebra of $\mathfrak{gl}
(V,V_*)$ as the centralizer of a maximal subalgebra every element of
which is semisimple, and then give the following description of all
Cartan subalgebras of $\mathfrak{gl} (V,V_*)$ under the assumption
that $\mathbb{K}$ is algebraically closed. A subalgebra of
$\mathfrak{gl} (V,V_*)$ is a Cartan subalgebra if and only if it
equals $\bigoplus_j \bigl( V_j \otimes (V_j)_* \bigr) \oplus (V^0 \otimes
V_*^0)$ for some onedimensional subspaces $V_j \subseteq V$ and
$(V_j)_* \subseteq V_*$ with $(V_i)_* (V_j) = \delta_{ij} \mathbb{K}$
and such that the spaces $V_*^0 = \bigcap_j (V_j)^\bot \subseteq V_*$
and $V^0 = \bigcap_j \bigl( (V_j)_* \bigr)^\bot \subseteq V$ satisfy
$V_*^0 (V^0) = \{0\}$. We then discuss explicit constructions of
subspaces $V_j$ and $(V_j)_*$ as above. Our second main result claims
that a Cartan subalgebra of $\mathfrak{gl} (V,V_*)$ can be described
alternatively as a locally nilpotent selfnormalizing subalgebra whose
adjoint representation is locally finite, or as a subalgebra
$\mathfrak{h}$ which coincides with the maximal locally nilpotent
$\mathfrak{h}$submodule of $\mathfrak{gl} (V,V_*)$, and such that the
adjoint representation of $\mathfrak{h}$ is locally finite.


617  On Harmonic Theory in Flows Pak, Hong Kyung
Recently [8], a harmonic theory was developed for a compact
contact manifold from the viewpoint of the transversal geometry of
contact flow. A contact flow is a typical example of geodesible
flow. As a natural generalization of the contact flow, the present
paper develops a harmonic theory for various flows on compact
manifolds. We introduce the notions of $H$harmonic and
$H^*$harmonic spaces associated to a H\"ormander flow. We also
introduce the notions of basic harmonic spaces associated to a weak
basic flow. One of our main results is to show that in the special
case of isometric flow these harmonic spaces are isomorphic to the
cohomology spaces of certain complexes. Moreover, we find an
obstruction for a geodesible flow to be isometric.


632  The Operator Amenability of Uniform Algebras Runde, Volker
We prove a quantized version of a theorem by M.~V.~She\u{\i}nberg:
A uniform algebra equipped with its canonical, {\it i.e.}, minimal,
operator space structure is operator amenable if and only if it is
a commutative $C^\ast$algebra.


635  Author Index  Index des auteurs 2003, for 2003  pour
No abstract.

