26. CJM 2011 (vol 64 pp. 573)
 Nawata, Norio

Fundamental Group of Simple $C^*$algebras with Unique Trace III
We introduce the fundamental group ${\mathcal F}(A)$ of
a simple $\sigma$unital $C^*$algebra $A$ with unique (up to scalar multiple)
densely defined lower semicontinuous trace.
This is a generalization of ``Fundamental Group of Simple
$C^*$algebras with Unique Trace I and II'' by Nawata and Watatani.
Our definition in this paper makes sense for stably projectionless $C^*$algebras.
We show that there exist separable stably projectionless $C^*$algebras such that
their fundamental groups are equal to $\mathbb{R}_+^\times$
by using the classification theorem of Razak and Tsang.
This is a contrast to the unital case in Nawata and Watatani.
This study is motivated by the work of Kishimoto and Kumjian.
Keywords:fundamental group, Picard group, Hilbert module, countable basis, stably projectionless algebra, dimension function Categories:46L05, 46L08, 46L35 

27. CJM 2011 (vol 64 pp. 455)
 Sherman, David

On Cardinal Invariants and Generators for von Neumann Algebras
We demonstrate how most common cardinal invariants associated with a von
Neumann algebra $\mathcal M$ can be computed from the decomposability number,
$\operatorname{dens}(\mathcal M)$, and the minimal cardinality of a generating
set, $\operatorname{gen}(\mathcal M)$.
Applications include the equivalence of the wellknown generator
problem, ``Is every separablyacting von Neumann algebra
singlygenerated?", with the formally stronger questions, ``Is every
countablygenerated von Neumann algebra singlygenerated?" and ``Is
the $\operatorname{gen}$ invariant monotone?" Modulo the generator problem, we
determine the range of the invariant $\bigl( \operatorname{gen}(\mathcal M),
\operatorname{dens}(\mathcal M) \bigr)$,
which is mostly governed by the inequality $\operatorname{dens}(\mathcal M) \leq
\mathfrak C^{\operatorname{gen}(\mathcal M)}$.
Keywords:von Neumann algebra, cardinal invariant, generator problem, decomposability number, representation density Category:46L10 

28. CJM 2011 (vol 63 pp. 798)
 Daws, Matthew

Representing Multipliers of the Fourier Algebra on NonCommutative $L^p$ Spaces
We show that the multiplier algebra of the Fourier algebra on a
locally compact group $G$ can be isometrically represented on a direct
sum on noncommutative $L^p$ spaces associated with the right von
Neumann algebra of $G$. The resulting image is the idealiser of the
image of the Fourier algebra. If these spaces are given their
canonical operator space structure, then we get a completely isometric
representation of the completely bounded multiplier algebra. We make
a careful study of the noncommutative $L^p$ spaces we construct and
show that they are completely isometric to those considered recently
by Forrest, Lee, and Samei. We improve a result of theirs about module
homomorphisms. We suggest a definition of a FigaTalamancaHerz
algebra built out of these noncommutative $L^p$ spaces, say
$A_p(\widehat G)$. It is shown that $A_2(\widehat G)$ is isometric to
$L^1(G)$, generalising the abelian situation.
Keywords:multiplier, Fourier algebra, noncommutative $L^p$ space, complex interpolation Categories:43A22, 43A30, 46L51, 22D25, 42B15, 46L07, 46L52 

29. CJM 2011 (vol 63 pp. 551)
 Hadwin, Don; Li, Qihui; Shen, Junhao

Topological Free Entropy Dimensions in Nuclear C$^*$algebras and in Full Free Products of Unital C$^*$algebras
In the paper, we introduce a new concept,
topological orbit dimension of an $n$tuple of elements in a unital
C$^{\ast}$algebra. Using this concept, we conclude that Voiculescu's
topological free
entropy dimension of every finite family of selfadjoint generators of a
nuclear C$^{\ast}$algebra is less than or equal to $1$. We also show that the
Voiculescu's topological free entropy dimension is additive in the full free
product of some unital C$^{\ast}$algebras. We show that the unital full free
product of Blackadar and Kirchberg's unital MF
algebras is also an MF algebra. As an application, we obtain that
$\mathop{\textrm{Ext}}(C_{r}^{\ast}(F_{2})\ast_{\mathbb{C}}C_{r}^{\ast}(F_{2}))$ is not a group.
Keywords: topological free entropy dimension, unital C$^{*}$algebra Categories:46L10, 46L54 

30. CJM 2011 (vol 63 pp. 381)
 Ji, Kui ; Jiang, Chunlan

A Complete Classification of AI Algebras with the Ideal Property
Let $A$ be an AI algebra; that is, $A$ is the $\mbox{C}^{*}$algebra inductive limit
of a sequence
$$
A_{1}\stackrel{\phi_{1,2}}{\longrightarrow}A_{2}\stackrel{\phi_{2,3}}{\longrightarrow}A_{3}
\longrightarrow\cdots\longrightarrow A_{n}\longrightarrow\cdots,
$$
where
$A_{n}=\bigoplus_{i=1}^{k_n}M_{[n,i]}(C(X^{i}_n))$,
$X^{i}_n$ are $[0,1]$, $k_n$, and
$[n,i]$ are positive integers.
Suppose that $A$ has the
ideal property: each closed twosided ideal of $A$ is generated by
the projections inside the ideal, as a closed twosided ideal.
In this article, we give a complete classification of AI algebras with the ideal property.
Keywords:AI algebras, Kgroup, tracial state, ideal property, classification Categories:46L35, 19K14, 46L05, 46L08 

31. CJM 2011 (vol 63 pp. 500)
 Dadarlat, Marius; Elliott, George A.; Niu, Zhuang

OneParameter Continuous Fields of Kirchberg Algebras. II
Parallel to the first two authors' earlier classification of separable, unita
oneparameter, continuous fields of Kirchberg algebras with torsion free
$\mathrm{K}$groups supported in one dimension, oneparameter, separable, uni
continuous fields of AFalgebras are classified by their ordered
$\mathrm{K}_0$sheaves. EffrosHandelmanShen type theorems are pr separable
unital oneparameter continuous fields of AFalgebras and Kirchberg algebras.
Keywords:continuous fields of C$^*$algebras, $\mathrm{K}_0$presheaves, EffrosHandeen type theorem Category:46L35 

32. CJM 2010 (vol 63 pp. 222)
 Wang, JiunChau

Limit Theorems for Additive Conditionally Free Convolution
In this paper we determine the limiting distributional behavior for
sums of infinitesimal conditionally free random variables. We show that the weak
convergence of classical convolution and that of conditionally free convolution
are equivalent for measures in an infinitesimal triangular array,
where the measures may have unbounded support. Moreover, we use these
limit theorems to study the conditionally free infinite divisibility. These results
are obtained by complex analytic methods without reference to the
combinatorics of cfree convolution.
Keywords:additive cfree convolution, limit theorems, infinitesimal arrays Categories:46L53, 60F05 

33. CJM 2010 (vol 63 pp. 3)
 Banica, T.; Belinschi, S. T.; Capitaine, M.; Collins, B.

Free Bessel Laws
We introduce and study a remarkable family of real probability
measures $\pi_{st}$ that we call free Bessel laws. These are related
to the free Poisson law $\pi$ via the formulae
$\pi_{s1}=\pi^{\boxtimes s}$ and ${\pi_{1t}=\pi^{\boxplus t}}$. Our
study includes definition and basic properties, analytic aspects
(supports, atoms, densities), combinatorial aspects (functional
transforms, moments, partitions), and a discussion of the relation
with random matrices and quantum groups.
Keywords:Poisson law, Bessel function, Wishart matrix, quantum group Categories:46L54, 15A52, 16W30 

34. CJM 2010 (vol 62 pp. 845)
 Samei, Ebrahim; Spronk, Nico; Stokke, Ross

Biflatness and PseudoAmenability of Segal Algebras
We investigate generalized amenability and biflatness properties of various (operator) Segal algebras in both the group algebra, $L^1(G)$, and the Fourier algebra, $A(G)$, of a locally compact group~$G$.
Keywords:Segal algebra, pseudoamenable Banach algebra, biflat Banach algebra Categories:43A20, 43A30, 46H25, 46H10, 46H20, 46L07 

35. CJM 2010 (vol 62 pp. 889)
 Xia, Jingbo

Singular Integral Operators and Essential Commutativity on the Sphere
Let ${\mathcal T}$ be the $C^\ast $algebra generated by the Toeplitz operators $\{T_\varphi : \varphi \in L^\infty (S,d\sigma )\}$ on the Hardy space $H^2(S)$ of the unit sphere in $\mathbf{C}^n$. It is well known that ${\mathcal T}$ is contained in the essential commutant of $\{T_\varphi : \varphi \in \operatorname{VMO}\cap L^\infty (S,d\sigma )\}$. We show that the essential commutant of $\{T_\varphi : \varphi \in \operatorname{VMO}\cap L^\infty (S,d\sigma )\}$ is strictly larger than ${\mathcal T}$.
Categories:32A55, 46L05, 47L80 

36. CJM 2009 (vol 61 pp. 1239)
 Davidson, Kenneth R.; Yang, Dilian

Periodicity in Rank 2 Graph Algebras
Kumjian and Pask introduced an aperiodicity condition
for higher rank graphs.
We present a detailed analysis of when this occurs
in certain rank 2 graphs.
When the algebra is aperiodic, we give another proof
of the simplicity of $\mathrm{C}^*(\mathbb{F}^+_{\theta})$.
The periodic $\mathrm{C}^*$algebras are characterized, and it is shown
that $\mathrm{C}^*(\mathbb{F}^+_{\theta}) \simeq
\mathrm{C}(\mathbb{T})\otimes\mathfrak{A}$
where $\mathfrak{A}$ is a simple $\mathrm{C}^*$algebra.
Keywords:higher rank graph, aperiodicity condition, simple $\mathrm{C}^*$algebra, expectation Categories:47L55, 47L30, 47L75, 46L05 

37. CJM 2009 (vol 61 pp. 1262)
 Dong, Z.

On the Local Lifting Properties of Operator Spaces
In this paper, we mainly study operator spaces which have the
locally lifting property (LLP). The dual of any ternary ring of operators is shown to
satisfy the strongly local reflexivity, and this is used to prove
that strongly local reflexivity holds also for operator spaces
which have the LLP. Several homological characterizations of the
LLP and weak expectation property are given. We also prove that for any operator space
$V$, $V^{**}$ has the LLP if and only if $V$ has the LLP and
$V^{*}$ is exact.
Keywords:operator space, locally lifting property, strongly locally reflexive Category:46L07 

38. CJM 2009 (vol 61 pp. 241)
 Azamov, N. A.; Carey, A. L.; Dodds, P. G.; Sukochev, F. A.

Operator Integrals, Spectral Shift, and Spectral Flow
We present a new and simple approach to the theory of multiple
operator integrals that applies to unbounded operators affiliated with general \vNa s.
For semifinite \vNa s we give applications
to the Fr\'echet differentiation of operator functions that sharpen existing results,
and establish the BirmanSolomyak representation of the spectral
shift function of M.\,G.\,Krein
in terms of an average of spectral measures in the type II setting.
We also exhibit a surprising connection between the spectral shift
function and spectral flow.
Categories:47A56, 47B49, 47A55, 46L51 

39. CJM 2008 (vol 60 pp. 975)
 Boca, Florin P.

An AF Algebra Associated with the Farey Tessellation
We associate with the Farey tessellation of the upper
halfplane an
AF algebra $\AA$ encoding the ``cutting sequences'' that define
vertical geodesics.
The EffrosShen AF algebras arise as quotients
of $\AA$. Using the path algebra model for AF algebras we construct, for
each $\tau \in \big(0,\frac{1}{4}\big]$, projections $(E_n)$ in
$\AA$ such that $E_n E_{n\pm 1}E_n \leq \tau E_n$.
Categories:46L05, 11A55, 11B57, 46L55, 37E05, 82B20 

40. CJM 2008 (vol 60 pp. 703)
 Toms, Andrew S.; Winter, Wilhelm

$\mathcal{Z}$Stable ASH Algebras
The JiangSu algebra $\mathcal{Z}$ has come to prominence in the
classification program for nuclear $C^*$algebras of late, due
primarily to the fact that Elliott's classification conjecture in its
strongest form predicts that all simple, separable, and nuclear
$C^*$algebras with unperforated $\mathrm{K}$theory will absorb
$\mathcal{Z}$ tensorially, i.e., will be $\mathcal{Z}$stable. There
exist counterexamples which suggest that the conjecture will only hold
for simple, nuclear, separable and $\mathcal{Z}$stable
$C^*$algebras. We prove that virtually all classes of nuclear
$C^*$algebras for which the Elliott conjecture has been confirmed so
far consist of $\mathcal{Z}$stable $C^*$algebras. This
follows in large part from the following result, also proved herein:
separable and approximately divisible $C^*$algebras are
$\mathcal{Z}$stable.
Keywords:nuclear $C^*$algebras, Ktheory, classification Categories:46L85, 46L35 

41. CJM 2008 (vol 60 pp. 189)
 Lin, Huaxin

Furstenberg Transformations and Approximate Conjugacy
Let $\alpha$ and
$\beta$ be two Furstenberg transformations on $2$torus associated
with irrational numbers $\theta_1,$ $\theta_2,$ integers $d_1, d_2$ and Lipschitz functions
$f_1$ and $f_2$. It is shown that $\alpha$ and $\beta$ are approximately conjugate in a
measure theoretical sense if (and only
if) $\overline{\theta_1\pm \theta_2}=0$ in $\R/\Z.$ Closely related to the classification of simple
amenable \CAs, it is shown that $\af$ and $\bt$ are approximately $K$conjugate if (and only if)
$\overline{\theta_1\pm \theta_2}=0$ in $\R/\Z$ and $d_1=d_2.$ This
is also shown to be equivalent to the condition that the associated crossed product \CAs are isomorphic.
Keywords:Furstenberg transformations, approximate conjugacy Categories:37A55, 46L35 

42. 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 

43. 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 

44. 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 

45. 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 

46. 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 

47. 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 

48. CJM 2005 (vol 57 pp. 983)
49. 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 

50. 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 
