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76. CJM 2006 (vol 58 pp. 768)
Decomposability of von Neumann Algebras and the Mazur Property of Higher Level The decomposability
number of a von Neumann algebra $\m$ (denoted by $\dec(\m)$) is the
greatest cardinality of a family of pairwise orthogonal nonzero
projections in $\m$. In this paper, we explore the close
connection between $\dec(\m)$ and the cardinal level of the Mazur
property for the predual $\m_*$ of $\m$, the study of which was
initiated by the second author. Here, our main focus is on
those von Neumann algebras whose preduals constitute such
important Banach algebras on a locally compact group $G$ as the
group algebra $\lone$, the Fourier algebra $A(G)$, the measure
algebra $M(G)$, the algebra $\luc^*$, etc. We show that for
any of these von Neumann algebras, say $\m$, the cardinal number
$\dec(\m)$ and a certain cardinal level of the Mazur property of $\m_*$
are completely encoded in the underlying group structure. In fact,
they can be expressed precisely by two dual cardinal
invariants of $G$: the compact covering number $\kg$ of $G$ and
the least cardinality $\bg$ of an open basis at the identity of
$G$. We also present an application of the Mazur property of higher
level to the topological centre problem for the Banach algebra
$\ag^{**}$.
Keywords:Mazur property, predual of a von Neumann algebra, locally compact group and its cardinal invariants, group algebra, Fourier algebra, topological centre Categories:22D05, 43A20, 43A30, 03E55, 46L10 
77. CJM 2006 (vol 58 pp. 859)
Nonstandard Ideals from Nonstandard Dual Pairs for $L^1(\omega)$ and $l^1(\omega)$ The Banach convolution algebras $l^1(\omega)$
and their continuous counterparts $L^1(\bR^+,\omega)$
are much
studied, because (when the submultiplicative weight function
$\omega$ is radical) they are pretty much the prototypic examples
of commutative radical Banach algebras. In cases of ``nice''
weights $\omega$, the only closed ideals they have are the obvious,
or ``standard'', ideals. But in the
general case, a brilliant but very difficult paper of Marc Thomas
shows that nonstandard ideals exist in $l^1(\omega)$. His
proof was successfully exported to the continuous case
$L^1(\bR^+,\omega)$ by Dales and McClure, but remained
difficult. In this paper we first present a small improvement: a
new and easier proof of the existence of nonstandard ideals in
$l^1(\omega)$ and $L^1(\bR^+,\omega)$. The new proof is based on
the idea of a ``nonstandard dual pair'' which we introduce.
We are then able to make a much larger improvement: we
find nonstandard ideals in $L^1(\bR^+,\omega)$ containing functions
whose supports extend all the way down to zero in $\bR^+$, thereby solving
what has become a notorious problem in the area.
Keywords:Banach algebra, radical, ideal, standard ideal, semigroup Categories:46J45, 46J20, 47A15 
78. CJM 2006 (vol 58 pp. 820)
Diametrically Maximal and Constant Width Sets in Banach Spaces We characterize diametrically maximal and constant width
sets in $C(K)$, where $K$ is any compact Hausdorff space. These
results are applied to prove that the sum of two diametrically
maximal sets needs not be diametrically maximal, thus solving a
question raised in a paper by Groemer. A~characterization of
diametrically maximal sets in $\ell_1^3$ is also given, providing
a negative answer to Groemer's problem in finite dimensional
spaces. We characterize constant width sets in $c_0(I)$, for
every $I$, and then we establish the connections between the Jung
constant of a Banach space and the existence of constant width
sets with empty interior. Porosity properties of families of sets
of constant width and rotundity properties of diametrically
maximal sets are also investigated. Finally, we present some
results concerning nonreflexive and Hilbert spaces.
Categories:52A05, 46B20 
79. CJM 2006 (vol 58 pp. 691)
Hypoelliptic BiInvariant Laplacians on Infinite Dimensional Compact Groups On a compact connected group $G$, consider the infinitesimal
generator $L$ of a central symmetric Gaussian convolution
semigroup $(\mu_t)_{t>0}$. Using appropriate notions of distribution
and smooth function spaces, we prove that $L$ is hypoelliptic if and only if
$(\mu_t)_{t>0} $ is absolutely continuous with respect to Haar measure
and admits a continuous density $x\mapsto \mu_t(x)$, $t>0$, such that
$\lim_{t\ra 0} t\log \mu_t(e)=0$. In particular, this condition holds
if and only if any Borel measure $u$ which is solution of $Lu=0$
in an open set $\Omega$ can be represented by a continuous
function in $\Omega$. Examples are discussed.
Categories:60B15, 43A77, 35H10, 46F25, 60J45, 60J60 
80. CJM 2006 (vol 58 pp. 492)
Extension Theorems on Weighted Sobolev Spaces and Some Applications We extend the extension theorems to weighted Sobolev spaces
$L^p_{w,k}(\mathcal D)$ on $(\varepsilon,\delta)$ domains with doubling weight $w$
that satisfies a Poincar\'e inequality and such that $w^{1/p}$ is locally
$L^{p'}$. We also make use of the main theorem to improve weighted
Sobolev interpolation inequalities.
Keywords:PoincarÃ© inequalities, $A_p$ weights, doubling weights, $(\ep,\delta)$ domain, $(\ep,\infty)$ domain Category:46E35 
81. CJM 2006 (vol 58 pp. 548)
Hausdorff and QuasiHausdorff Matrices on Spaces of Analytic Functions We consider Hausdorff and quasiHausdorff matrices as operators
on classical spaces of analytic functions such as the Hardy and
the Bergman spaces, the Dirichlet space, the Bloch spaces and $\BMOA$. When the generating
sequence of the matrix is the moment sequence of a measure $\mu$,
we find the conditions on $\mu$ which are equivalent to the boundedness
of the matrix on the various spaces.
Categories:47B38, 46E15, 40G05, 42A20 
82. CJM 2006 (vol 58 pp. 39)
$C^*$Algebras of Irreversible Dynamical Systems We show that certain $C^*$algebras which have been studied by,
among others, Arzumanian, Vershik, Deaconu, and Renault, in
connection with a measurepreserving transformation of a measure space
or a covering map of a compact space, are special cases of the
endomorphism crossedproduct construction recently introduced by the
first named author. As a consequence these algebras are given
presentations in terms of generators and relations. These results
come as a consequence of a general theorem on faithfulness of
representations which are covariant with respect to certain circle
actions. For the case of topologically free covering maps we prove a
stronger result on faithfulness of representations which needs no
covariance. We also give a necessary and sufficient condition for
simplicity.
Categories:46L55, 37A55 
83. CJM 2005 (vol 57 pp. 1249)
Strictly Singular and Cosingular Multiplications Let $L(X)$ be the space of bounded linear operators on the Banach space $X$.
We study the strict singularity andcosingularity of the twosided multiplication
operators $S \mapsto ASB$ on $L(X)$, where $A,B \in L(X)$ are fixed bounded
operators and $X$ is a classical Banach space. Let $1

84. CJM 2005 (vol 57 pp. 1056)
Hyperbolic Group $C^*$Algebras and FreeProduct $C^*$Algebras as Compact Quantum Metric Spaces Let $\ell$ be a length function on a group $G$, and let $M_{\ell}$
denote the
operator of pointwise multiplication by $\ell$ on $\bell^2(G)$.
Following Connes,
$M_{\ell}$ can be used as a ``Dirac'' operator for $C_r^*(G)$. It defines a
Lipschitz seminorm on $C_r^*(G)$, which defines a metric on the state space of
$C_r^*(G)$. We show that if $G$ is a hyperbolic group and if $\ell$ is
a wordlength function on $G$, then the topology from this metric
coincides with the
weak$*$ topology (our definition of a ``compact quantum metric
space''). We show that a convenient framework is that of filtered
$C^*$algebras which satisfy a suitable ``Haageruptype'' condition. We
also use this
framework to prove an analogous fact for certain reduced
free products of $C^*$algebras.
Categories:46L87, 20F67, 46L09 
85. CJM 2005 (vol 57 pp. 983)
A Symmetric Imprimitivity Theorem for Commuting Proper Actions We prove a symmetric imprimitivity theorem for commuting proper
actions of locally compact groups $H$ and $K$ on a $C^*$algebra.
Categories:46L05, 46L08, 46L55 
86. CJM 2005 (vol 57 pp. 897)
Representation of Banach Ideal Spaces and Factorization of Operators Representation theorems are proved for Banach ideal spaces with the Fatou property
which are built by the Calder{\'o}nLozanovski\u\i\ construction.
Factorization theorems for operators in spaces more general than the Lebesgue
$L^{p}$ spaces are investigated. It is natural to extend the Gagliardo
theorem on the Schur test and the Rubio de~Francia theorem on factorization of the
Muckenhoupt $A_{p}$ weights to reflexive Orlicz spaces. However, it turns out that for
the scales far from $L^{p}$spaces this is impossible. For the concrete integral operators
it is shown that factorization theorems and the Schur test in some reflexive Orlicz spaces
are not valid. Representation theorems for the Calder{\'o}nLozanovski\u\i\ construction
are involved in the proofs.
Keywords:Banach ideal spaces, weighted spaces, weight functions,, CalderÃ³nLozanovski\u\i\ spaces, Orlicz spaces, representation of, spaces, uniqueness problem, positive linear operators, positive sublinear, operators, Schur test, factorization of operators, f Categories:46E30, 46B42, 46B70 
87. CJM 2005 (vol 57 pp. 673)
On the Structure of the Spreading Models of a Banach Space We study some questions concerning the structure of the
set of spreading models of a separable infinitedimensional Banach
space $X$. In particular we give an example of a reflexive $X$ so that
all spreading models of $X$ contain $\ell_1$ but none of them is
isomorphic to $\ell_1$. We also prove that for any countable set $C$
of spreading models generated by weakly null sequences there is a
spreading model generated by a weakly null sequence which dominates
each element of $C$. In certain cases this ensures that $X$ admits,
for each $\alpha < \omega_1$, a spreading model $(\tilde
x_i^{(\alpha)})_i$ such that if $\alpha < \beta$ then $(\tilde
x_i^{(\alpha)})_i$ is dominated by (and not equivalent to)
$(\tilde x_i^{(\beta)})_i$. Some applications of these ideas are used to
give sufficient conditions on a Banach space for the existence of a
subspace and an operator defined on the subspace, which is not a
compact perturbation of a multiple of the inclusion map.
Category:46B03 
88. CJM 2005 (vol 57 pp. 351)
Extensions by Simple $C^*$Algebras: Quasidiagonal Extensions Let $A$ be an amenable separable $C^*$algebra and $B$ be a nonunital
but $\sigma$unital simple $C^*$algebra with continuous scale.
We show that two essential extensions
$\tau_1$ and $\tau_2$ of $A$ by $B$ are approximately
unitarily equivalent if and only if
$$
[\tau_1]=[\tau_2] \text{ in } KL(A, M(B)/B).
$$
If $A$ is assumed to satisfy the Universal Coefficient Theorem,
there is a bijection from approximate unitary equivalence
classes of the above mentioned extensions to
$KL(A, M(B)/B)$.
Using $KL(A, M(B)/B)$, we compute exactly when an essential extension
is quasidiagonal. We show that quasidiagonal extensions
may not be approximately trivial.
We also study the approximately trivial extensions.
Keywords:Extensions, Simple $C^*$algebras Categories:46L05, 46L35 
89. CJM 2005 (vol 57 pp. 17)
On Amenability and CoAmenability of Algebraic Quantum Groups and Their Corepresentations We introduce and study several notions of amenability for unitary
corepresentations and $*$representations of algebraic quantum groups,
which may be used to characterize amenability and coamenability for
such quantum groups. As a background for this study, we investigate
the associated tensor C$^{*}$categories.
Keywords:quantum group, amenability Categories:46L05, 46L65, 22D10, 22D25, 43A07, 43A65, 58B32 
90. CJM 2005 (vol 57 pp. 61)
On Operators with Spectral Square but without Resolvent Points Decompositions of spectral type are
obtained for closed Hilbert space operators with empty resolvent
set, but whose square has closure which is spectral. Krein space
situations are also discussed.
Keywords:unbounded operators, closed operators,, spectral resolution, indefinite metric Categories:47A05, 47A15, 47B40, 47B50, 46C20 
91. CJM 2004 (vol 56 pp. 1237)
Central Sequence Algebras of a Purely Infinite Simple $C^{*}$algebra We are concerned with a unital separable nuclear purely infinite
simple $C^{*}$algebra\ $A$ satisfying UCT with a Rohlin flow, as a
continuation of~\cite{Kismh}. Our first result (which is
independent of the Rohlin flow) is to characterize when two {\em
central} projections in $A$ are equivalent by a {\em central}
partial isometry. Our second result shows that the Ktheory of
the central sequence algebra $A'\cap A^\omega$ (for an $\omega\in
\beta\N\setminus\N$) and its {\em fixed point} algebra under the
flow are the same (incorporating the previous result). We will
also complete and supplement the characterization result of the
Rohlin property for flows stated in~ \cite{Kismh}.
Category:46L40 
92. CJM 2004 (vol 56 pp. 1121)
Division par un polynÃ´me hyperbolique On se donne un intervalle ouvert non vide $\omega$ de
$\mathbb R$, un ouvert connexe non vide $\Omega$ de $\mathbb R_s$ et
un polyn\^ome unitaire
\[
P_m(z, \lambda) = z^m + a_1(\lambda)z^{m1} = +\dots + a_{m1}(\lambda)
z + a_m(\lambda),
\]
de degr\'e $m>0$, d\'ependant du param\`etre $\lambda \in \Omega$. Un
tel polyn\^ome est dit $\omega$hyperbolique si, pour tout $\lambda
\in \Omega$, ses racines sont r\'eelles et appartiennent \`a $\omega$.
On suppose que les fonctions $a_k, \, k=1, \dots, m$, appartiennent \`a
une classe ultradiff\'erentiable $C_M(\Omega)$. On s`int\'eresse au
probl\`eme suivant. Soit $f$ appartient \`a $C_M(\Omega)$, existetil
des fonctions $Q_f$ et $R_{f,k},\, k=0, \dots, m1$, appartenant
respectivement \`a $C_M(\omega \times \Omega)$ et \`a $C_M(\Omega)$,
telles que l'on ait, pour $(x,\lambda) \in \omega \times \Omega$,
\[
f(x) = P_m(x,\lambda) Q_f (x,\lambda) + \sum^{m1}_{k=0} x^k
R_{f,k}(\lambda)~?
\]
On donne ici une r\'eponse positive d\`es que le polyn\^ome est
$\omega$hyperbolique, que la class untradiff\'eren\tiable soit
quasianalytique ou non ; on obtient alors, des exemples d'id\'eaux
ferm\'es dans $C_M(\mathbb R^n)$. On compl\`ete ce travail par une
g\'en\'eralisation d'un r\'esultat de C.~L. Childress dans le cadre
quasianalytique et quelques remarques.
Categories:26E10, 46E25, 46J20 
93. CJM 2004 (vol 56 pp. 983)
Fubini's Theorem for Ultraproducts \\of Noncommutative $L_p$Spaces Let $(\M_i)_{i\in I}$, $(\N_j)_{j\in J}$ be families of von
Neumann algebras and $\U$, $\U'$ be ultrafilters in $I$, $J$,
respectively. Let $1\le p<\infty$ and $\nen$. Let $x_1$,\dots,$x_n$ in
$\prod L_p(\M_i)$ and $y_1$,\dots,$y_n$ in $\prod L_p(\N_j)$ be
bounded families. We show the following equality
$$
\lim_{i,\U} \lim_{j,\U'} \Big\ \summ_{k=1}^n x_k(i)\otimes
y_k(j)\Big\_{L_p(\M_i\otimes \N_j)} = \lim_{j,\U'} \lim_{i,\U}
\Big\ \summ_{k=1}^n x_k(i)\otimes y_k(j)\Big\_{L_p(\M_i\otimes \N_j)} .
$$
For $p=1$ this Fubini type result is related to the local
reflexivity of duals of $C^*$algebras. This fails for $p=\infty$.
Keywords:noncommutative $L_p$spaces, ultraproducts Categories:46L52, 46B08, 46L07 
94. CJM 2004 (vol 56 pp. 926)
KHomology of the Rotation Algebras $A_{\theta}$ We study the Khomology of the rotation algebras
$A_{\theta}$ using the sixterm cyclic sequence
for the Khomology of a crossed product by
${\bf Z}$. In the case that $\theta$ is irrational,
we use Pimsner and Voiculescu's work on AFembeddings
of the $A_{\theta}$ to search for the missing
generator of the even Khomology.
Categories:58B34, 19K33, 46L 
95. CJM 2004 (vol 56 pp. 843)
Type Decomposition and the Rectangular AFD Property for $W^*$TRO's We study the type decomposition and the rectangular AFD property for
$W^*$TRO's. Like von Neumann algebras, every $W^*$TRO can be
uniquely decomposed into the direct sum of $W^*$TRO's of
type $I$, type $II$, and type $III$.
We may further consider $W^*$TRO's of type $I_{m, n}$
with cardinal numbers $m$ and $n$, and consider $W^*$TRO's of
type $II_{\lambda, \mu}$ with $\lambda, \mu = 1$ or $\infty$.
It is shown that every separable stable $W^*$TRO
(which includes type $I_{\infty,\infty}$, type $II_{\infty,
\infty}$ and type $III$) is TROisomorphic to a von Neumann algebra.
We also introduce the rectangular version of the approximately finite
dimensional property for $W^*$TRO's.
One of our major results is to show that a separable $W^*$TRO
is injective if and only
if it is rectangularly approximately finite dimensional.
As a consequence of this result, we show that a dual operator space
is injective if and only if its operator predual is a rigid
rectangular ${\OL}_{1, 1^+}$ space (equivalently, a rectangular
Categories:46L07, 46L08, 46L89 
96. CJM 2004 (vol 56 pp. 699)
Bump Functions with HÃ¶lder Derivatives We study the range of the gradients
of a $C^{1,\al}$smooth bump function defined on a Banach space.
We find that this set must satisfy two geometrical conditions:
It can not be too flat and it satisfies a strong compactness condition
with respect to an appropriate distance.
These notions are defined precisely below.
With these results we illustrate the differences with
the case of $C^1$smooth bump functions.
Finally, we give a sufficient condition on a subset of $X^{\ast}$ so that it is
the set of the gradients of a $C^{1,1}$smooth bump function.
In particular, if $X$ is an infinite dimensional Banach space
with a $C^{1,1}$smooth bump function,
then any convex open bounded subset of $X^{\ast}$ containing $0$ is the set
of the gradients of a $C^{1,1}$smooth bump function.
Keywords:Banach space, bump function, range of the derivative Categories:46T20, 26E15, 26B05 
97. CJM 2004 (vol 56 pp. 472)
InfiniteDimensional Polyhedrality This paper deals with generalizations of the notion of a polytope to infinite
dimensions. The most general definition is the following: a bounded closed
convex subset of a Banach space is called a \emph{polytope} if each of its
finitedimensional affine sections is a (standard) polytope.
We study the relationships between eight known definitions
of infinitedimensional
polyhedrality. We provide a complete isometric
classification of them, which gives
solutions to several open problems.
An almost complete isomorphic classification
is given as well (only one implication remains open).
Categories:46B20, 46B03, 46B04, 52B99 
98. CJM 2004 (vol 56 pp. 225)
Complex Uniform Convexity and Riesz Measure The norm on a Banach space gives rise to a subharmonic function on the
complex plane for which the distributional Laplacian gives a Riesz measure.
This measure is calculated explicitly here for Lebesgue $L^p$ spaces and the
von~NeumannSchatten trace ideals. Banach spaces that are $q$uniformly
$\PL$convex in the sense of Davis, Garling and TomczakJaegermann are
characterized in terms of the mass distribution of this measure. This gives
a new proof that the trace ideals $c^p$ are $2$uniformly $\PL$convex for
$1\leq p\leq 2$.
Keywords:subharmonic functions, Banach spaces, Schatten trace ideals Categories:46B20, 46L52 
99. CJM 2004 (vol 56 pp. 3)
Locally Compact Pro$C^*$Algebras Let $X$ be a locally compact noncompact Hausdorff topological space. Consider
the algebras $C(X)$, $C_b(X)$, $C_0(X)$, and $C_{00}(X)$ of respectively arbitrary,
bounded, vanishing at infinity, and compactly supported continuous functions on $X$.
Of these, the second and third are $C^*$algebras, the fourth is a normed algebra,
whereas the first is only a topological algebra (it is indeed a pro$C^\ast$algebra).
The interesting fact about these algebras is that if one of them is given, the
others can be obtained using functional analysis tools. For instance, given the
$C^\ast$algebra $C_0(X)$, one can get the other three algebras by
$C_{00}(X)=K\bigl(C_0(X)\bigr)$, $C_b(X)=M\bigl(C_0(X)\bigr)$, $C(X)=\Gamma\bigl(
K(C_0(X))\bigr)$, where the right hand sides are the Pedersen ideal, the
multiplier algebra, and the unbounded multiplier algebra of the Pedersen ideal of
$C_0(X)$, respectively. In this article we consider the possibility of these
transitions for general $C^\ast$algebras. The difficult part is to start with a
pro$C^\ast$algebra $A$ and to construct a $C^\ast$algebra $A_0$ such that
$A=\Gamma\bigl(K(A_0)\bigr)$. The pro$C^\ast$algebras for which this is
possible are called {\it locally compact\/} and we have characterized them using
a concept similar to that of an approximate identity.
Keywords:pro$C^\ast$algebras, projective limit, multipliers of Pedersen's ideal Categories:46L05, 46M40 
100. CJM 2003 (vol 55 pp. 1302)
The Ideal Structures of Crossed Products of Cuntz Algebras by QuasiFree Actions of Abelian Groups We completely determine the ideal structures of the crossed
products of Cuntz algebras by quasifree actions of abelian groups
and give another proof of A.~Kishimoto's result on the simplicity
of such crossed products. We also give a necessary and sufficient
condition that our algebras become primitive, and compute the
Connes spectra and $K$groups of our algebras.
Categories:46L05, 46L55, 46L45 