101. CJM 2005 (vol 57 pp. 983)
102. CJM 2005 (vol 57 pp. 1056)
 Ozawa, Narutaka; Rieffel, Marc A.

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 

103. CJM 2005 (vol 57 pp. 897)
 Berezhnoĭ, Evgenii I.; Maligranda, Lech

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 

104. CJM 2005 (vol 57 pp. 673)
 Androulakis, G.; Odell, E.; Schlumprecht, Th.; TomczakJaegermann, N.

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 

105. CJM 2005 (vol 57 pp. 351)
 Lin, Huaxin

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 

106. CJM 2005 (vol 57 pp. 61)
 Binding, Paul; Strauss, Vladimir

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 

107. CJM 2005 (vol 57 pp. 17)
 Bédos, Erik; Conti, Roberto; Tuset, Lars

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 

108. CJM 2004 (vol 56 pp. 1237)
 Kishimoto, Akitaka

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 

109. CJM 2004 (vol 56 pp. 1121)
 Chaumat, Jacques; Chollet, AnneMarie

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 

110. CJM 2004 (vol 56 pp. 983)
 Junge, Marius

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 

111. CJM 2004 (vol 56 pp. 926)
 Hadfield, Tom

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 

112. 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 

113. 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 

114. 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 

115. 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 

116. 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 

117. CJM 2003 (vol 55 pp. 1302)
118. 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 

119. 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 

120. 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 

121. 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 

122. 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 

123. 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 

124. 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 

125. 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 
