1. CJM 2017 (vol 70 pp. 294)
 Eilers, Søren; Restorff, Gunnar; Ruiz, Efren; Sørensen, Adam P. W.

Geometric classification of graph C*algebras over finite graphs
We address the classification problem for graph $C^*$algebras of
finite graphs (finitely many edges and vertices), containing
the class of CuntzKrieger algebras as a
prominent special case. Contrasting earlier work, we do not assume
that the graphs satisfy the standard condition (K), so that the
graph
$C^*$algebras may come with uncountably many ideals.
We find that in this generality, stable isomorphism of graph
$C^*$algebras does not coincide with the geometric notion of Cuntz
move equivalence. However, adding a modest condition on the
graphs, the two notions are proved to be mutually equivalent and
equivalent to the $C^*$algebras having isomorphic $K$theories. This
proves in turn that under this condition, the graph
$C^*$algebras are in fact classifiable by $K$theory, providing in
particular complete classification when the $C^*$algebras in question
are either of real rank zero or type I/postliminal. The key ingredient
in obtaining these results is a characterization of Cuntz move
equivalence using the adjacency matrices of the graphs.
Our results are applied to discuss the classification problem
for the quantum lens spaces defined by Hong and SzymaÅski,
and to complete the classification of graph $C^*$algebras associated to
all simple graphs with four vertices or less.
Keywords:graph $C^*$algebra, geometric classification, $K$theory, flow equivalence Categories:46L35, 46L80, 46L55, 37B10 

2. CJM 2017 (vol 70 pp. 26)
 Bosa, Joan; Petzka, Henning

Comparison Properties of the Cuntz semigroup and applications to C*algebras
We study comparison properties in the category $\mathrm{Cu}$ aiming to
lift results to the C*algebraic setting. We introduce a new
comparison property and relate it to both the CFP and $\omega$comparison.
We show differences of all properties by providing examples,
which suggest that the corona factorization for C*algebras might
allow for both finite and infinite projections. In addition,
we show that R{\o}rdam's simple, nuclear C*algebra with a finite
and an infinite projection does not have the CFP.
Keywords:classification of C*algebras, cuntz semigroup Categories:46L35, 06F05, 46L05, 19K14 

3. CJM 2014 (vol 67 pp. 55)
 Barron, Tatyana; Kerner, Dmitry; Tvalavadze, Marina

On Varieties of Lie Algebras of Maximal Class
We study complex projective varieties that parametrize
(finitedimensional) filiform Lie algebras over ${\mathbb C}$,
using equations derived by Millionshchikov. In the
infinitedimensional case we concentrate our attention on
${\mathbb N}$graded Lie algebras of maximal class. As shown by A.
Fialowski
there are only
three isomorphism types of $\mathbb{N}$graded Lie algebras
$L=\oplus^{\infty}_{i=1} L_i$ of maximal class generated by $L_1$
and $L_2$, $L=\langle L_1, L_2 \rangle$. Vergne described the
structure of these algebras with the property $L=\langle L_1
\rangle$. In this paper we study those generated by the first and
$q$th components where $q\gt 2$, $L=\langle L_1, L_q \rangle$. Under
some technical condition, there can only be one isomorphism type
of such algebras. For $q=3$ we fully classify them. This gives a
partial answer to a question posed by Millionshchikov.
Keywords:filiform Lie algebras, graded Lie algebras, projective varieties, topology, classification Categories:17B70, 14F45 

4. CJM 2013 (vol 66 pp. 596)
 Eilers, Søren; Restorff, Gunnar; Ruiz, Efren

The Ordered $K$theory of a Full Extension
Let $\mathfrak{A}$ be a $C^{*}$algebra with real rank zero which has
the stable weak cancellation property. Let $\mathfrak{I}$ be an ideal
of $\mathfrak{A}$ such that $\mathfrak{I}$ is stable and satisfies the
corona factorization property. We prove that
$
0 \to \mathfrak{I} \to \mathfrak{A} \to \mathfrak{A} / \mathfrak{I} \to 0
$
is a full extension if and only if the extension is stenotic and
$K$lexicographic. {As an immediate application, we extend the
classification result for graph $C^*$algebras obtained by Tomforde
and the first named author to the general nonunital case. In
combination with recent results by Katsura, Tomforde, West and the
first author, our result may also be used to give a purely
$K$theoretical description of when an essential extension of two
simple and stable graph $C^*$algebras is again a graph
$C^*$algebra.}
Keywords:classification, extensions, graph algebras Categories:46L80, 46L35, 46L05 

5. 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 

6. CJM 2009 (vol 62 pp. 305)
 Hua, He; Yunbai, Dong; Xianzhou, Guo

Approximation and Similarity Classification of Stably Finitely Strongly Irreducible Decomposable Operators
Let $\mathcal H$ be a complex separable Hilbert space and ${\mathcal L}({\mathcal H})$ denote the collection of bounded linear operators on ${\mathcal H}$. In this paper, we show that for any operator $A\in{\mathcal L}({\mathcal H})$, there exists a stably finitely (SI) decomposable operator $A_\epsilon$, such that $\AA_{\epsilon}\<\epsilon$ and ${\mathcal{\mathcal A}'(A_{\epsilon})}/\operatorname{rad} {{\mathcal A}'(A_{\epsilon})}$ is commutative, where $\operatorname{rad}{{\mathcal A}'(A_{\epsilon})}$ is the Jacobson radical of ${{\mathcal A}'(A_{\epsilon})}$. Moreover, we give a similarity classification of the stably finitely decomposable operators that generalizes the result on similarity classification of CowenDouglas operators given by C. L. Jiang.
Keywords:$K_{0}$group, strongly irreducible decomposition, CowenâDouglas operators, commutant algebra, similarity classification Categories:47A05, 47A55, 46H20 

7. 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 

8. 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 

9. CJM 1997 (vol 49 pp. 675)
 de Cataldo, Mark Andrea A.

Some adjunctiontheoretic properties of codimension two nonsingular subvarities of quadrics
We make precise the structure of the first two reduction morphisms
associated with codimension two nonsingular subvarieties
of nonsingular quadrics $\Q^n$, $n\geq 5$.
We give a coarse classification of the same class of subvarieties
when they are assumed not to be of loggeneraltype.}
Keywords:Adjunction Theory, classification, codimension two, conic bundles,, low codimension, non loggeneraltype, quadric, reduction, special, variety. Categories:14C05, 14E05, 14E25, 14E30, 14E35, 14J10 
