26. 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 

27. 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 

28. 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 

29. CJM 2003 (vol 55 pp. 1302)
30. 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 

31. 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 

32. 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 

33. 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 

34. 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 

35. CJM 2001 (vol 53 pp. 161)
 Lin, Huaxin

Classification of Simple Tracially AF $C^*$Algebras
We prove that preclassifiable (see 3.1) simple nuclear tracially AF
\CA s (TAF) are classified by their $K$theory. As a consequence all
simple, locally AH and TAF \CA s are in fact AH algebras (it is known
that there are locally AH algebras that are not AH). We also prove
the following Rationalization Theorem. Let $A$ and $B$ be two unital
separable nuclear simple TAF \CA s with unique normalized traces
satisfying the Universal Coefficient Theorem. If $A$ and $B$ have the
same (ordered and scaled) $K$theory and $K_0 (A)_+$ is locally
finitely generated, then $A \otimes Q \cong B \otimes Q$, where $Q$ is
the UHFalgebra with the rational $K_0$. Classification results (with
restriction on $K_0$theory) for the above \CA s are also obtained.
For example, we show that, if $A$ and $B$ are unital nuclear separable
simple TAF \CA s with the unique normalized trace satisfying the UCT
and with $K_1(A) = K_1(B)$, and $A$ and $B$ have the same rational
(scaled ordered) $K_0$, then $A \cong B$. Similar results are also
obtained for some cases in which $K_0$ is nondivisible such as
$K_0(A) = \mathbf{Z} [1/2]$.
Categories:46L05, 46L35 

36. CJM 1999 (vol 51 pp. 850)
 Muhly, Paul S.; Solel, Baruch

Tensor Algebras, Induced Representations, and the Wold Decomposition
Our objective in this sequel to \cite{MSp96a} is to develop extensions,
to representations of tensor algebras over $C^{*}$correspondences, of
two fundamental facts about isometries on Hilbert space: The Wold
decomposition theorem and Beurling's theorem, and to apply these to
the analysis of the invariant subspace structure of certain subalgebras
of CuntzKrieger algebras.
Keywords:tensor algebras, correspondence, induced representation, Wold decomposition, Beurling's theorem Categories:46L05, 46L40, 46L89, 47D15, 47D25, 46M10, 46M99, 47A20, 47A45, 47B35 

37. CJM 1998 (vol 50 pp. 323)
 Dykema, Kenneth J.; Rørdam, Mikael

Purely infinite, simple $C^\ast$algebras arising from free product constructions
Examples of simple, separable, unital, purely infinite
$C^\ast$algebras are constructed, including:
\item{(1)} some that are not approximately divisible;
\item{(2)} those that arise as crossed products of any of a certain class of
$C^\ast$algebras by any of a certain class of nonunital endomorphisms;
\item{(3)} those that arise as reduced free products of pairs of
$C^\ast$algebras with respect to any from a certain class of states.
Categories:46L05, 46L45 

38. CJM 1997 (vol 49 pp. 1188)
 Leen, Michael J.

Factorization in the invertible group of a $C^*$algebra
In this paper we consider the following problem:
Given a unital \cs\ $A$ and a collection of elements $S$ in the
identity component of the invertible group of $A$, denoted \ino,
characterize the group of finite products of elements of $S$. The
particular $C^*$algebras studied in this paper are either
unital purely infinite simple or of the form \tenp, where $A$ is
any \cs\ and $K$ is the compact operators on an infinite dimensional
separable Hilbert space. The types of elements used in the
factorizations are unipotents ($1+$ nilpotent), positive invertibles
and symmetries ($s^2=1$). First we determine the groups of finite
products for each collection of elements in \tenp. Then we give
upper bounds on the number of factors needed in these cases. The main
result, which uses results for \tenp, is that for $A$ unital purely
infinite and simple, \ino\ is generated by each of these collections
of elements.
Category:46L05 

39. CJM 1997 (vol 49 pp. 963)
 Lin, Huaxin

Homomorphisms from $C(X)$ into $C^*$algebras
Let $A$ be a simple $C^*$algebra
with real rank zero, stable rank one and weakly
unperforated $K_0(A)$ of countable rank. We show that
a monomorphism $\phi\colon C(S^2) \to A$ can be approximated
pointwise by homomorphisms from $C(S^2)$ into $A$ with
finite dimensional range if and only if certain index
vanishes. In particular, we show that every homomorphism
$\phi$ from $C(S^2)$ into a UHFalgebra can be approximated
pointwise by homomorphisms from $C(S^2)$ into the UHFalgebra
with finite dimensional range. As an application, we show
that if $A$ is a simple $C^*$algebra of real rank zero
and is an inductive limit of matrices over $C(S^2)$ then
$A$ is an AFalgebra. Similar results for tori are also
obtained. Classification of ${\bf Hom}\bigl(C(X),A\bigr)$
for lower dimensional spaces is also studied.
Keywords:Homomorphism of $C(S^2)$, approximation, real, rank zero, classification Categories:46L05, 46L80, 46L35 
