51. CJM 2007 (vol 59 pp. 966)
 Forrest, Brian E.; Runde, Volker; Spronk, Nico

Operator Amenability of the Fourier Algebra in the $\cb$Multiplier Norm
Let $G$ be a locally compact group, and let $A_{\cb}(G)$ denote the
closure of $A(G)$, the Fourier algebra of $G$, in the space of completely
bounded multipliers of $A(G)$. If $G$ is a weakly amenable, discrete group
such that $\cstar(G)$ is residually finitedimensional, we show that
$A_{\cb}(G)$ is operator amenable. In particular,
$A_{\cb}(\free_2)$ is operator amenable even though $\free_2$, the free
group in two generators, is not an amenable group. Moreover, we show that
if $G$ is a discrete group such that $A_{\cb}(G)$ is operator amenable,
a closed ideal of $A(G)$ is weakly completely complemented in $A(G)$
if and only if it has an approximate identity bounded in the $\cb$multiplier
norm.
Keywords:$\cb$multiplier norm, Fourier algebra, operator amenability, weak amenability Categories:43A22, 43A30, 46H25, 46J10, 46J40, 46L07, 47L25 

52. CJM 2007 (vol 59 pp. 343)
 Lin, Huaxin

Weak Semiprojectivity in Purely Infinite Simple $C^*$Algebras
Let $A$ be a separable amenable purely infinite simple \CA which
satisfies the Universal Coefficient Theorem. We prove that $A$ is
weakly semiprojective if and only if $K_i(A)$ is a countable
direct sum of finitely generated groups ($i=0,1$). Therefore, if
$A$ is such a \CA, for any $\ep>0$ and any finite subset ${\mathcal
F}\subset A$ there exist $\dt>0$ and a finite subset ${\mathcal
G}\subset A$ satisfying the following: for any contractive
positive linear map $L: A\to B$ (for any \CA $B$) with $
\L(ab)L(a)L(b)\<\dt$ for $a, b\in {\mathcal G}$
there exists a homomorphism $h\from A\to B$ such that
$ \h(a)L(a)\<\ep$ for $a\in {\mathcal F}$.
Keywords:weakly semiprojective, purely infinite simple $C^*$algebras Categories:46L05, 46L80 

53. CJM 2006 (vol 58 pp. 1268)
 Sims, Aidan

GaugeInvariant Ideals in the $C^*$Algebras of Finitely Aligned HigherRank Graphs
We produce a complete description of the lattice of gaugeinvariant
ideals in $C^*(\Lambda)$ for a finitely aligned $k$graph
$\Lambda$. We provide a condition on $\Lambda$ under which every ideal
is gaugeinvariant. We give conditions on $\Lambda$ under which
$C^*(\Lambda)$ satisfies the hypotheses of the KirchbergPhillips
classification theorem.
Keywords:Graphs as categories, graph algebra, $C^*$algebra Category:46L05 

54. CJM 2006 (vol 58 pp. 1144)
 Hamana, Masamichi

Partial $*$Automorphisms, Normalizers, and Submodules in Monotone Complete $C^*$Algebras
For monotone complete $C^*$algebras
$A\subset B$ with $A$ contained in $B$ as a monotone closed
$C^*$subalgebra, the relation $X = AsA$
gives a bijection between the set of all
monotone closed linear subspaces $X$ of $B$ such that
$AX + XA \subset X$
and
$XX^* + X^*X \subset A$
and a set of certain partial
isometries $s$ in the ``normalizer" of $A$ in $B$,
and similarly for the map $s \mapsto \Ad s$
between the latter set and a set of certain ``partial $*$automorphisms"
of $A$.
We introduce natural inverse semigroup
structures in the set of such $X$'s and the set of
partial $*$automorphisms of $A$, modulo a certain relation, so that
the composition of these maps induces an inverse semigroup
homomorphism between them.
For a large enough $B$ the homomorphism becomes surjective and
all the partial $*$automorphisms of
$A$ are realized via partial isometries in $B$.
In particular, the inverse semigroup associated with
a type ${\rm II}_1$ von Neumann factor,
modulo the outer automorphism group,
can be viewed as the fundamental group of the factor.
We also consider the $C^*$algebra version of these results.
Categories:46L05, 46L08, 46L40, 20M18 

55. CJM 2006 (vol 58 pp. 768)
 Hu, Zhiguo; Neufang, Matthias

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 

56. CJM 2006 (vol 58 pp. 39)
 Exel, R.; Vershik, A.

$C^*$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 

57. CJM 2005 (vol 57 pp. 983)
58. 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 

59. 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 

60. 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 

61. 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 

62. 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 

63. 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 

64. 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 

65. 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 

66. 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 

67. CJM 2003 (vol 55 pp. 1302)
68. 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 

69. 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 

70. 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 

71. 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 

72. 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 

73. 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 

74. 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 

75. 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 
