101. CJM 2004 (vol 56 pp. 699)
 Gaspari, Thierry

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 

102. CJM 2004 (vol 56 pp. 843)
 Ruan, ZhongJin

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 

103. CJM 2004 (vol 56 pp. 472)
 Fonf, Vladimir P.; Veselý, Libor

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 

104. CJM 2004 (vol 56 pp. 225)
 Blower, Gordon; Ransford, Thomas

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 

105. CJM 2004 (vol 56 pp. 3)
 Amini, Massoud

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 

106. CJM 2003 (vol 55 pp. 1302)
107. CJM 2003 (vol 55 pp. 969)
 Glöckner, Helge

Lie Groups of Measurable Mappings
We describe new construction principles for infinitedimensional Lie
groups. In particular, given any measure space $(X,\Sigma,\mu)$ and
(possibly infinitedimensional) Lie group $G$, we construct a Lie
group $L^\infty (X,G)$, which is a Fr\'echetLie group if $G$ is so.
We also show that the weak direct product $\prod^*_{i\in I} G_i$ of an
arbitrary family $(G_i)_{i\in I}$ of Lie groups can be made a Lie
group, modelled on the locally convex direct sum $\bigoplus_{i\in I}
L(G_i)$.
Categories:22E65, 46E40, 46E30, 22E67, 46T20, 46T25 

108. CJM 2003 (vol 55 pp. 204)
 Yan, Yaqiang

On the Nonsquare Constants of Orlicz Spaces with Orlicz Norm
Let $l^{\Phi}$ and $L^\Phi (\Omega)$ be the Orlicz sequence space and
function space generated by $N$function $\Phi(u)$ with Orlicz norm.
We give equivalent expressions for the nonsquare constants $C_J
(l^\Phi)$, $C_J \bigl( L^\Phi (\Omega) \bigr)$ in sense of James and
$C_S (l^\Phi)$, $C_S \bigl( L^\Phi(\Omega) \bigr)$ in sense of
Sch\"affer. We are devoted to get practical computational formulas
giving estimates of these constants and to obtain their exact value in
a class of spaces $l^{\Phi}$ and $L^\Phi (\Omega)$.
Keywords:James nonsquare constant, SchÃ¤ffer nonsquare constant, Orlicz sequence space, Orlicz function space Category:46E30 

109. CJM 2002 (vol 54 pp. 1280)
 Skrzypczak, Leszek

Besov Spaces and Hausdorff Dimension For Some CarnotCarathÃ©odory Metric Spaces
We regard a system of left invariant vector fields $\mathcal{X}=\{X_1,\dots,X_k\}$
satisfying the H\"ormander condition and the related CarnotCarath\'eodory metric on a
unimodular Lie group $G$. We define Besov spaces corresponding to the subLaplacian
$\Delta=\sum X_i^2$ both with positive and negative smoothness. The atomic
decomposition of the spaces is given. In consequence we get the distributional
characterization of the Hausdorff dimension of Borel subsets with the Haar measure
zero.
Keywords:Besov spaces, subelliptic operators, CarnotCarathÃ©odory metric, Hausdorff dimension Categories:46E35, 43A15, 28A78 

110. CJM 2002 (vol 54 pp. 1165)
 Blasco, Oscar; Arregui, José Luis

Multipliers on Vector Valued Bergman Spaces
Let $X$ be a complex Banach space and let $B_p(X)$ denote the
vectorvalued Bergman space on the unit disc for $1\le p<\infty$. A
sequence $(T_n)_n$ of bounded operators between two Banach spaces $X$
and $Y$ defines a multiplier between $B_p(X)$ and $B_q(Y)$
(resp.\ $B_p(X)$ and $\ell_q(Y)$) if for any function $f(z) =
\sum_{n=0}^\infty x_n z^n$ in $B_p(X)$ we have that $g(z) =
\sum_{n=0}^\infty T_n (x_n) z^n$ belongs to $B_q(Y)$ (resp.\
$\bigl( T_n (x_n) \bigr)_n \in \ell_q(Y)$). Several results on these
multipliers are obtained, some of them depending upon the Fourier or
Rademacher type of the spaces $X$ and $Y$. New properties defined by
the vectorvalued version of certain inequalities for Taylor
coefficients of functions in $B_p(X)$ are introduced.
Categories:42A45, 46E40 

111. CJM 2002 (vol 54 pp. 1100)
 Wood, Peter J.

The Operator Biprojectivity of the Fourier Algebra
In this paper, we investigate projectivity in the category of operator
spaces. In particular, we show that the Fourier algebra of a locally
compact group $G$ is operator biprojective if and only if $G$ is
discrete.
Keywords:locally compact group, Fourier algebra, operator space, projective Categories:13D03, 18G25, 43A95, 46L07, 22D99 

112. CJM 2002 (vol 54 pp. 694)
 Gabriel, Michael J.

Cuntz Algebra States Defined by Implementers of Endomorphisms of the $\CAR$ Algebra
We investigate representations of the Cuntz algebra $\mathcal{O}_2$
on antisymmetric Fock space $F_a (\mathcal{K}_1)$ defined by
isometric implementers of certain quasifree endomorphisms of the
CAR algebra in pure quasifree states $\varphi_{P_1}$. We pay
corresponding to these representations and the Fock special
attention to the vector states on $\mathcal{O}_2$ vacuum, for which
we obtain explicit formulae. Restricting these states to the
gaugeinvariant subalgebra $\mathcal{F}_2$, we find that for
natural choices of implementers, they are again pure quasifree and
are, in fact, essentially the states $\varphi_{P_1}$. We proceed to
consider the case for an arbitrary pair of implementers, and deduce
that these Cuntz algebra representations are irreducible, as are their
restrictions to $\mathcal{F}_2$.
The endomorphisms of $B \bigl( F_a (\mathcal{K}_1) \bigr)$ associated
with these representations of $\mathcal{O}_2$ are also considered.
Categories:46L05, 46L30 

113. CJM 2002 (vol 54 pp. 634)
 Weber, Eric

Frames and Single Wavelets for Unitary Groups
We consider a unitary representation of a discrete countable abelian
group on a separable Hilbert space which is associated to a cyclic
generalized frame multiresolution analysis. We extend Robertson's
theorem to apply to frames generated by the action of the group.
Within this setup we use Stone's theorem and the theory of projection
valued measures to analyze wandering frame collections. This yields a
functional analytic method of constructing a wavelet from a
generalized frame multi\resolution analysis in terms of the frame
scaling vectors. We then explicitly apply our results to the action
of the integers given by translations on $L^2({\mathbb R})$.
Keywords:wavelet, multiresolution analysis, unitary group representation, frame Categories:42C40, 43A25, 42C15, 46N99 

114. CJM 2002 (vol 54 pp. 225)
115. CJM 2002 (vol 54 pp. 303)
 Ghahramani, Fereidoun; Grabiner, Sandy

Convergence Factors and Compactness in Weighted Convolution Algebras
We study convergence in weighted convolution algebras $L^1(\omega)$ on
$R^+$, with the weights chosen such that the corresponding weighted
space $M(\omega)$ of measures is also a Banach algebra and is the dual
space of a natural related space of continuous functions. We
determine convergence factor $\eta$ for which
weak$^\ast$convergence of $\{\lambda_n\}$ to $\lambda$ in $M(\omega)$
implies norm convergence of $\lambda_n \ast f$ to $\lambda \ast f$ in
$L^1 (\omega\eta)$. We find necessary and sufficent conditions which
depend on $\omega$ and $f$ and also find necessary and sufficent
conditions for $\eta$ to be a convergence factor for all $L^1(\omega)$
and all $f$ in $L^1(\omega)$. We also give some applications to the
structure of weighted convolution algebras. As a preliminary result
we observe that $\eta$ is a convergence factor for $\omega$ and $f$ if
and only if convolution by $f$ is a compact operator from $M(\omega)$
(or $L^1(\omega)$) to $L^1 (\omega\eta)$.
Categories:43A10, 43A15, 46J45, 46J99 

116. CJM 2002 (vol 54 pp. 138)
 Razak, Shaloub

On the Classification of Simple Stably Projectionless $\C^*$Algebras
It is shown that simple stably projectionless $\C^S*$algebras which
are inductive limits of certain specified building blocks with trivial
$\K$theory are classified by their cone of positive traces with
distinguished subset. This is the first example of an isomorphism
theorem verifying the conjecture of Elliott for a subclass of the
stably projectionless algebras.
Categories:46L35, 46L05 

117. CJM 2001 (vol 53 pp. 1223)
 Mygind, Jesper

Classification of Certain Simple $C^*$Algebras with Torsion in $K_1$
We show that the Elliott invariant is a classifying invariant for the
class of $C^*$algebras that are simple unital infinite dimensional
inductive limits of finite direct sums of building blocks of the form
$$
\{f \in C(\T) \otimes M_n : f(x_i) \in M_{d_i}, i = 1,2,\dots,N\},
$$
where $x_1,x_2,\dots,x_N \in \T$, $d_1,d_2,\dots,d_N$ are integers
dividing $n$, and $M_{d_i}$ is embedded unitally into $M_n$.
Furthermore we prove existence and uniqueness theorems for
$*$homomorphisms between such algebras and we identify the range of
the invariant.
Categories:46L80, 19K14, 46L05 

118. CJM 2001 (vol 53 pp. 979)
 Nagisa, Masaru; Osaka, Hiroyuki; Phillips, N. Christopher

Ranks of Algebras of Continuous $C^*$Algebra Valued Functions
We prove a number of results about the stable and particularly the
real ranks of tensor products of \ca s under the assumption that one
of the factors is commutative. In particular, we prove the following:
{\raggedright
\begin{enumerate}[(5)]
\item[(1)] If $X$ is any locally compact $\sm$compact Hausdorff space
and $A$ is any \ca, then\break
$\RR \bigl( C_0 (X) \otimes A \bigr) \leq
\dim (X) + \RR(A)$.
\item[(2)] If $X$ is any locally compact Hausdorff space and $A$ is
any \pisca, then $\RR \bigl( C_0 (X) \otimes A \bigr) \leq 1$.
\item[(3)] $\RR \bigl( C ([0,1]) \otimes A \bigr) \geq 1$ for any
nonzero \ca\ $A$, and $\sr \bigl( C ([0,1]^2) \otimes A \bigr) \geq 2$
for any unital \ca\ $A$.
\item[(4)] If $A$ is a unital \ca\ such that $\RR(A) = 0$, $\sr (A) =
1$, and $K_1 (A) = 0$, then\break
$\sr \bigl( C ([0,1]) \otimes A \bigr) = 1$.
\item[(5)] There is a simple separable unital nuclear \ca\ $A$ such
that $\RR(A) = 1$ and\break
$\sr \bigl( C ([0,1]) \otimes A \bigr) = 1$.
\end{enumerate}}
Categories:46L05, 46L52, 46L80, 19A13, 19B10 

119. CJM 2001 (vol 53 pp. 1031)
 Sampson, G.; Szeptycki, P.

The Complete $(L^p,L^p)$ Mapping Properties of Some Oscillatory Integrals in Several Dimensions
We prove that the operators $\int_{\mathbb{R}_+^2} e^{ix^a \cdot
y^b} \varphi (x,y) f(y)\, dy$ map $L^p(\mathbb{R}^2)$ into itself
for $p \in J =\bigl[\frac{a_l+b_l}{a_l+(\frac{b_l r}{2})},\frac{a_l+b_l}
{a_l(1\frac{r}{2})}\bigr]$ if $a_l,b_l\ge 1$ and $\varphi(x,y)=xy^{r}$,
$0\le r <2$, the result is sharp. Generalizations to dimensions $d>2$
are indicated.
Categories:42B20, 46B70, 47G10 

120. CJM 2001 (vol 53 pp. 809)
 Robertson, Guyan; Steger, Tim

Asymptotic $K$Theory for Groups Acting on $\tA_2$ Buildings
Let $\Gamma$ be a torsion free lattice in $G=\PGL(3, \mathbb{F})$ where
$\mathbb{F}$ is a nonarchimedean local field. Then $\Gamma$ acts freely
on the affine BruhatTits building $\mathcal{B}$ of $G$ and there is an
induced action on the boundary $\Omega$ of $\mathcal{B}$. The crossed
product $C^*$algebra $\mathcal{A}(\Gamma)=C(\Omega) \rtimes \Gamma$
depends only on $\Gamma$ and is classified by its $K$theory. This article
shows how to compute the $K$theory of $\mathcal{A}(\Gamma)$ and of the
larger class of rank two CuntzKrieger algebras.
Keywords:$K$theory, $C^*$algebra, affine building Categories:46L80, 51E24 

121. CJM 2001 (vol 53 pp. 565)
 Hare, Kathryn E.; Sato, Enji

Spaces of Lorentz Multipliers
We study when the spaces of Lorentz multipliers from $L^{p,t}
\rightarrow L^{p,s}$ are distinct. Our main interest is the case when
$s
Keywords:multipliers, convolution operators, Lorentz spaces, Lorentzimproving multipliers Categories:43A22, 42A45, 46E30 

122. CJM 2001 (vol 53 pp. 592)
 Perera, Francesc

Ideal Structure of Multiplier Algebras of Simple $C^*$algebras With Real Rank Zero
We give a description of the monoid of Murrayvon Neumann equivalence
classes of projections for multiplier algebras of a wide class of
$\sigma$unital simple $C^\ast$algebras $A$ with real rank zero and stable
rank one. The lattice of ideals of this monoid, which is known to be
crucial for understanding the ideal structure of the multiplier
algebra $\mul$, is therefore analyzed. In important cases it is shown
that, if $A$ has finite scale then the quotient of $\mul$ modulo any
closed ideal $I$ that properly contains $A$ has stable rank one. The
intricacy of the ideal structure of $\mul$ is reflected in the fact
that $\mul$ can have uncountably many different quotients, each one
having uncountably many closed ideals forming a chain with respect to
inclusion.
Keywords:$C^\ast$algebra, multiplier algebra, real rank zero, stable rank, refinement monoid Categories:46L05, 46L80, 06F05 

123. CJM 2001 (vol 53 pp. 631)
 Walters, Samuel G.

KTheory of NonCommutative Spheres Arising from the Fourier Automorphism
For a dense $G_\delta$ set of real parameters $\theta$ in $[0,1]$
(containing the rationals) it is shown that the group $K_0 (A_\theta
\rtimes_\sigma \mathbb{Z}_4)$ is isomorphic to $\mathbb{Z}^9$, where
$A_\theta$ is the rotation C*algebra generated by unitaries $U$, $V$
satisfying $VU = e^{2\pi i\theta} UV$ and $\sigma$ is the Fourier
automorphism of $A_\theta$ defined by $\sigma(U) = V$, $\sigma(V) =
U^{1}$. More precisely, an explicit basis for $K_0$ consisting of
nine canonical modules is given. (A slight generalization of this
result is also obtained for certain separable continuous fields of
unital C*algebras over $[0,1]$.) The Connes Chern character $\ch
\colon K_0 (A_\theta \rtimes_\sigma \mathbb{Z}_4) \to H^{\ev} (A_\theta
\rtimes_\sigma \mathbb{Z}_4)^*$ is shown to be injective for a dense
$G_\delta$ set of parameters $\theta$. The main computational tool in
this paper is a group homomorphism $\vtr \colon K_0 (A_\theta
\rtimes_\sigma \mathbb{Z}_4) \to \mathbb{R}^8 \times \mathbb{Z}$
obtained from the Connes Chern character by restricting the
functionals in its codomain to a certain ninedimensional subspace of
$H^{\ev} (A_\theta \rtimes_\sigma \mathbb{Z}_4)$. The range of $\vtr$
is fully determined for each $\theta$. (We conjecture that this
subspace is all of $H^{\ev}$.)
Keywords:C*algebras, Ktheory, automorphisms, rotation algebras, unbounded traces, Chern characters Categories:46L80, 46L40, 19K14 

124. CJM 2001 (vol 53 pp. 546)
 Erlijman, Juliana

MultiSided Braid Type Subfactors
We generalise the twosided construction of examples of pairs of
subfactors of the hyperfinite II$_1$ factor $R$ in [E1]which arise
by considering unitary braid representations with certain
propertiesto multisided pairs. We show that the index for the
multisided pair can be expressed as a power of that for the
twosided pair. This construction can be applied to the natural
exampleswhere the braid representations are obtained in connection
with the representation theory of Lie algebras of types $A$, $B$, $C$,
$D$. We also compute the (first) relative commutants.
Category:46L37 

125. CJM 2001 (vol 53 pp. 325)
 Matui, Hiroki

Ext and OrderExt Classes of Certain Automorphisms of $C^*$Algebras Arising from Cantor Minimal Systems
Giordano, Putnam and Skau showed that the transformation group
$C^*$algebra arising from a Cantor minimal system is an $AT$algebra,
and classified it by its $K$theory. For approximately inner
automorphisms that preserve $C(X)$, we will determine their classes in
the Ext and OrderExt groups, and introduce a new invariant for the
closure of the topological full group. We will also prove that every
automorphism in the kernel of the homomorphism into the Ext group is
homotopic to an inner automorphism, which extends Kishimoto's result.
Categories:46L40, 46L80, 54H20 
