Search: MSC category 46L
( Selfadjoint operator algebras ($C^$algebras, von Neumann ($W^*$) algebras, etc.) [See also 22D25, 47Lxx] *$algebras, von Neumann ($W^*$) algebras, etc.) [See also 22D25, 47Lxx] * )
1. CJM Online first
 BenaychGeorges, Florent; Cébron, Guillaume; Rochet, Jean

Fluctuation of matrix entries and application to outliers of elliptic matrices
For any family of $N\times N$ random matrices $(\mathbf{A}_k)_{k\in
K}$ which is invariant, in law, under unitary conjugation, we
give general sufficient conditions for central limit theorems
for random variables of the type $\operatorname{Tr}(\mathbf{A}_k
\mathbf{M})$, where the matrix $\mathbf{M}$ is deterministic
(such random variables include for example the normalized matrix
entries of the $\mathbf{A}_k$'s). A consequence is the asymptotic
independence of the projection of the matrices $\mathbf{A}_k$
onto the subspace of null trace matrices from their projections
onto the orthogonal of this subspace. These results are used
to study the asymptotic behavior of the outliers of a spiked
elliptic random matrix. More precisely, we show that the fluctuations
of these outliers around their limits can have various rates
of convergence, depending on the Jordan Canonical Form of the
additive perturbation. Also, some correlations can arise between
outliers at a macroscopic distance from each other. These phenomena
have already been observed
with random matrices
from the Single Ring Theorem.
Keywords:random matrix, Gaussian fluctuation, spiked model, elliptic random matrix, Weingarten calculus, Haar measure Categories:60B20, 15B52, 60F05, 46L54 

2. CJM Online first
 Courtney, Kristin; Shulman, Tatiana

Elements of $C^*$algebras attaining their norm in a finitedimensional representation
We characterize the class of RFD $C^*$algebras as those containing
a dense subset of elements that attain their norm under a finitedimensional
representation. We show further that this subset is the whole
space precisely when every irreducible representation of the
$C^*$algebra is finitedimensional, which is equivalent to the
$C^*$algebra having no simple infinitedimensional AF subquotient.
We apply techniques from this proof to show the existence of
elements in more general classes of $C^*$algebras whose norms
in finitedimensional representations fit certain prescribed
properties.
Keywords:AFtelescope, RFD, projective Categories:46L05, 47A67 

3. CJM Online first
 Osaka, Hiroyuki; Teruya, Tamotsu

The JiangSu absorption for inclusions of unital C*algebras
We introduce the tracial Rokhlin property for a conditional expectation
for an inclusion
of unital C*algebras $P \subset A$ with index finite, and show
that an action $\alpha$
from a finite group $G$ on a simple unital C*algebra $A$ has
the tracial Rokhlin property
in the sense of N. C. Phillips
if and only if the canonical conditional expectation $E\colon
A \rightarrow A^G$ has the tracial
Rokhlin property.
Let $\mathcal{C}$ be a class of infinite dimensional stably
finite separable unital C*algebras
which is closed under the following conditions:
(1)
If $A \in {\mathcal C}$ and $B \cong A$, then $B \in \mathcal{C}$.
(2)
If $A \in \mathcal{C}$ and $n \in \mathbb{N}$, then $M_n(A) \in \mathcal{C}$.
(3)
If $A \in \mathcal{C}$ and $p \in A$ is a nonzero projection,
then $pAp \in \mathcal{C}$.
Suppose that any C*algebra in $\mathcal{C}$ is weakly semiprojective.
We prove that if $A$ is a local tracial $\mathcal{C}$algebra
in the sense of Fan and Fang and a conditional expectation
$E\colon A \rightarrow P$ is of indexfinite type with the tracial
Rokhlin property, then $P$ is a unital
local tracial $\mathcal{C}$algebra.
The main result is that if $A$ is simple, separable, unital
nuclear, JiangSu absorbing
and $E\colon A \rightarrow P$ has the tracial Rokhlin property,
then $P$ is JiangSu absorbing.
As an application, when an action $\alpha$
from a finite group $G$ on a simple unital C*algebra $A$ has
the tracial Rokhlin property,
then for any subgroup $H$ of $G$ the fixed point algebra $A^H$
and the crossed product algebra
$A \rtimes_{\alpha_{H}} H$ is JiangSu absorbing.
We also show that the strict comparison property for a Cuntz
semigroup $W(A)$
is hereditary to $W(P)$ if $A$ is simple, separable, exact,
unital,
and $E\colon A \rightarrow P$ has the tracial Rokhlin property.
Keywords:JiangSu absorption, inclusion of C*algebra, strict comparison Categories:46L55, 46L35 

4. CJM Online first
 Li, Hui; Yang, Dilian

Boundary quotient C*algebras of products of odometers
In this paper, we study the boundary quotient C*algebras associated
to products of odometers. One of our main results
shows that the boundary quotient C*algebra of the standard product
of $k$ odometers
over $n_i$letter alphabets ($1\le i\le k$) is always nuclear,
and that
it is a UCT Kirchberg algebra
if and only if $\{\ln n_i: 1\le i\le k\}$ is rationally independent,
if and only if the associated singlevertex $k$graph C*algebra
is simple.
To achieve this, one of our main steps is to construct a topological
$k$graph such that
its associated CuntzPimsner C*algebra is isomorphic to the
boundary quotient C*algebra.
Some relations between the boundary quotient C*algebra and the
C*algebra $\mathrm{Q}_\mathbb{N}$ introduced by Cuntz are also
investigated.
Keywords:C*algebra; semigroup; odometer; topological $k$graph; product system; ZappaSzÃ©p product Category:46L05 

5. CJM Online first
 Georgescu, Magdalena Cecilia

Integral Formula for Spectral Flow for $p$Summable Operators
Fix a von Neumann algebra $\mathcal{N}$ equipped with a suitable trace
$\tau$. For a path of selfadjoint BreuerFredholm operators, the
spectral flow measures the net amount of spectrum which moves from
negative to nonnegative. We consider specifically the case of paths
of bounded perturbations of a fixed unbounded selfadjoint
BreuerFredholm operator affiliated with $\mathcal{N}$. If the unbounded
operator is psummable (that is, its resolvents are contained in the
ideal $L^p$), then it is possible to obtain an integral formula which
calculates spectral flow. This integral formula was first proven by
Carey and Phillips, building on earlier approaches of Phillips. Their
proof was based on first obtaining a formula for the larger class of
$\theta$summable operators, and then using Laplace transforms to
obtain a psummable formula. In this paper, we present a direct proof
of the psummable formula, which is both shorter and simpler than
theirs.
Keywords:spectral flow, $p$summable Fredholm module Categories:19k56, 46L87, , 58B34 

6. CJM Online first
 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 

7. CJM 2017 (vol 69 pp. 1385)
 Pasnicu, Cornel; Phillips, N. Christopher

The Weak Ideal Property and Topological Dimension Zero
Following up on previous work,
we prove a number of results for C*algebras
with the weak ideal property
or topological dimension zero,
and some results for C*algebras with related properties.
Some of the more important results include:
$\bullet$
The weak ideal property
implies topological dimension zero.
$\bullet$
For a separable C*algebra~$A$,
topological dimension zero is equivalent to
${\operatorname{RR}} ({\mathcal{O}}_2 \otimes A) = 0$,
to $D \otimes A$ having the ideal property
for some (or any) Kirchberg algebra~$D$,
and to $A$ being residually hereditarily in
the class of all C*algebras $B$ such that
${\mathcal{O}}_{\infty} \otimes B$
contains a nonzero projection.
$\bullet$
Extending the known result for ${\mathbb{Z}}_2$,
the classes of C*algebras
with residual (SP),
which are residually hereditarily (properly) infinite,
or which are purely infinite and have the ideal property,
are closed under crossed products by arbitrary actions
of abelian $2$groups.
$\bullet$
If $A$ and $B$ are separable,
one of them is exact,
$A$ has the ideal property,
and $B$ has the weak ideal property,
then $A \otimes_{\mathrm{min}} B$ has the weak ideal property.
$\bullet$
If $X$ is a totally disconnected locally compact Hausdorff space
and $A$ is a $C_0 (X)$algebra
all of whose fibers have one of the weak ideal property,
topological dimension zero,
residual (SP),
or the combination of pure infiniteness and the ideal property,
then $A$ also has the corresponding property
(for topological dimension zero, provided $A$ is separable).
$\bullet$
Topological dimension zero,
the weak ideal property,
and the ideal property
are all equivalent
for a substantial class of separable C*algebras including
all separable locally AH~algebras.
$\bullet$
The weak ideal property does not imply the ideal property
for separable $Z$stable C*algebras.
We give other related results,
as well as counterexamples to several other statements
one might hope for.
Keywords:ideal property, weak ideal property, topological dimension zero, $C_0 (X)$algebra, purely infinite C*algebra Category:46L05 

8. CJM Online first
 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 

9. CJM 2017 (vol 69 pp. 1109)
 Ng, P. W.; Skoufranis, P.

Closed Convex Hulls of Unitary Orbits in Certain Simple Real Rank Zero C$^*$algebras
In this paper, we characterize the closures of convex hulls of
unitary orbits of selfadjoint operators in unital, separable,
simple C$^*$algebras with nontrivial tracial simplex, real
rank zero, stable rank one, and strict comparison of projections
with respect to tracial states. In addition, an upper bound
for the number of unitary conjugates in a convex combination
needed to approximate a selfadjoint are obtained.
Keywords:convex hull of unitary orbits, real rank zero C*algebras simple, eigenvalue function, majorization Category:46L05 

10. CJM 2017 (vol 69 pp. 1064)
 Crann, Jason

Amenability and Covariant Injectivity of Locally Compact Quantum Groups II
Building on our previous work, we study the nonrelative homology
of quantum group convolution algebras. Our main result establishes
the equivalence of amenability of a locally compact quantum group
$\mathbb{G}$ and 1injectivity of
$L^{\infty}(\widehat{\mathbb{G}})$
as an operator
$L^1(\widehat{\mathbb{G}})$module.
In particular, a locally compact group $G$ is amenable if and
only if its group von Neumann algebra
$VN(G)$
is 1injective as
an operator module over the Fourier algebra $A(G)$. As an application,
we provide a decomposability result for completely bounded
$L^1(\widehat{\mathbb{G}})$module
maps on
$L^{\infty}(\widehat{\mathbb{G}})$,
and give a simplified proof that amenable discrete
quantum groups have coamenable compact duals which avoids the
use of modular theory and the PowersStÃ¸rmer inequality, suggesting
that our homological techniques may yield a new approach to the
open problem of duality between amenability and coamenability.
Keywords:locally compact quantum group, amenability, injective module Categories:22D35, 46M10, 46L89 

11. CJM 2016 (vol 69 pp. 548)
 Hartglass, Michael

Free Product C*algebras Associated with Graphs, Free Differentials, and Laws of Loops
We study a canonical C$^*$algebra, $\mathcal{S}(\Gamma, \mu)$, that
arises from a weighted graph $(\Gamma, \mu)$, specific cases
of which were previously studied in the context of planar algebras.
We discuss necessary and sufficient conditions of the weighting
which ensure simplicity and uniqueness of trace of $\mathcal{S}(\Gamma,
\mu)$, and study the structure of its positive cone. We then
study the $*$algebra, $\mathcal{A}$, generated by the generators of
$\mathcal{S}(\Gamma, \mu)$, and use a free differential calculus and
techniques of Charlesworth and Shlyakhtenko, as well as Mai,
Speicher, and Weber to show that certain ``loop" elements have
no atoms in their spectral measure. After modifying techniques
of Shlyakhtenko and Skoufranis to show that self adjoint elements
$x \in M_{n}(\mathcal{A})$ have algebraic Cauchy transform, we explore
some applications to eigenvalues of polynomials in Wishart matrices
and to diagrammatic elements in von Neumann algebras initially
considered by Guionnet, Jones, and Shlyakhtenko.
Keywords:free probability, C*algebra Category:46L09 

12. CJM 2016 (vol 68 pp. 999)
 Izumi, Masaki; Morrison, Scott; Penneys, David

Quotients of $A_2 * T_2$
We study unitary quotients of the free product unitary pivotal
category $A_2*T_2$.
We show that such quotients are parametrized by an integer $n\geq
1$ and an $2n$th root of unity $\omega$.
We show that for $n=1,2,3$, there is exactly one quotient and
$\omega=1$.
For $4\leq n\leq 10$, we show that there are no such quotients.
Our methods also apply to quotients of $T_2*T_2$, where we have
a similar result.
The essence of our method is a consistency check on jellyfish
relations.
While we only treat the specific cases of $A_2 * T_2$ and $T_2
* T_2$, we anticipate that our technique can be extended to a
general method for proving nonexistence of planar algebras with
a specified principal graph.
During the preparation of this manuscript, we learnt of Liu's
independent result on composites of $A_3$ and $A_4$ subfactor
planar algebras
(arxiv:1308.5691).
In 1994, BischHaagerup showed that the principal graph of a
composite of $A_3$ and $A_4$ must fit into a certain family,
and Liu has classified all such subfactor planar algebras.
We explain the connection between the quotient categories and
the corresponding composite subfactor planar algebras.
As a corollary of Liu's result, there are no such quotient categories
for $n\geq 4$.
This is an abridged version of
arxiv:1308.5723.
Keywords:pivotal category, free product, quotient, subfactor, intermediate subfactor Category:46L37 

13. CJM 2016 (vol 69 pp. 373)
 Kaftal, Victor; Ng, Ping Wong; Zhang, Shuang

Strict Comparison of Positive Elements in Multiplier Algebras
Main result: If a C*algebra $\mathcal{A}$ is simple, $\sigma$unital,
has finitely many extremal traces, and has strict comparison
of positive elements by traces, then its multiplier algebra
$\operatorname{\mathcal{M}}(\mathcal{A})$
also has strict comparison of positive elements by traces. The
same results holds if ``finitely many extremal traces" is replaced
by ``quasicontinuous scale".
A key ingredient in the proof is that every positive element
in the multiplier algebra of an arbitrary $\sigma$unital C*algebra
can be approximated by a bidiagonal series.
An application of strict comparison: If $\mathcal{A}$ is a simple separable
stable C*algebra with real rank zero, stable rank one, and
strict comparison of positive elements by traces, then whether
a positive element is a positive linear combination of projections
is determined by the trace values of its range projection.
Keywords:strict comparison, bidiagonal form, positive combinations Categories:46L05, 46L35, 46L45, 47C15 

14. CJM 2016 (vol 68 pp. 1023)
 Phillips, John; Raeburn, Iain

Centrevalued Index for Toeplitz Operators with Noncommuting Symbols
We formulate and prove a ``winding number'' index
theorem for certain ``Toeplitz'' operators in the same spirit
as GohbergKrein, Lesch and others. The ``number'' is replaced
by a selfadjoint operator in a subalgebra $Z\subseteq Z(A)$
of a unital $C^*$algebra, $A$. We assume a faithful $Z$valued
trace $\tau$ on $A$ left invariant under an action $\alpha:{\mathbf
R}\to Aut(A)$ leaving $Z$ pointwise fixed.If $\delta$ is the
infinitesimal generator of $\alpha$ and $u$ is invertible in
$\operatorname{dom}(\delta)$ then the
``winding operator'' of $u$ is $\frac{1}{2\pi i}\tau(\delta(u)u^{1})\in
Z_{sa}.$ By a careful choice of representations we extend $(A,Z,\tau,\alpha)$
to a von Neumann setting
$(\mathfrak{A},\mathfrak{Z},\bar\tau,\bar\alpha)$ where $\mathfrak{A}=A^{\prime\prime}$
and $\mathfrak{Z}=Z^{\prime\prime}.$
Then $A\subset\mathfrak{A}\subset \mathfrak{A}\rtimes{\bf R}$, the von
Neumann crossed product, and there is a faithful, dual $\mathfrak{Z}$trace
on $\mathfrak{A}\rtimes{\bf R}$. If $P$ is the projection in $\mathfrak{A}\rtimes{\bf
R}$
corresponding to the nonnegative spectrum of the generator of
$\mathbf R$ inside $\mathfrak{A}\rtimes{\mathbf R}$ and
$\tilde\pi:A\to\mathfrak{A}\rtimes{\mathbf R}$
is the embedding then we define for $u\in A^{1}$, $T_u=P\tilde\pi(u)
P$
and show it is Fredholm in an appropriate sense and the $\mathfrak{Z}$valued
index of $T_u$ is the negative of the winding operator.
In outline the proof follows the proof of the scalar case done
previously by the authors. The main difficulty is making sense
of the constructions with the scalars replaced by $\mathfrak{Z}$ in
the von Neumann setting. The construction of the dual $\mathfrak{Z}$trace
on $\mathfrak{A}\rtimes{\mathbf R}$ required the nontrivial development
of a $\mathfrak{Z}$Hilbert Algebra theory. We show that certain of
these Fredholm operators fiber as a ``section'' of Fredholm operators
with scalarvalued index and the centrevalued index fibers as
a section of the scalarvalued indices.
Keywords:index ,Toeplitz operator Categories:46L55, 19K56, 46L80 

15. CJM 2016 (vol 68 pp. 1067)
 Runde, Volker; Viselter, Ami

On Positive Definiteness over Locally Compact Quantum Groups
The notion of positivedefinite functions over locally compact
quantum
groups was recently introduced and studied by Daws and Salmi.
Based
on this work, we generalize various wellknown results about
positivedefinite
functions over groups to the quantum framework. Among these are
theorems
on "square roots" of positivedefinite functions, comparison
of
various topologies, positivedefinite measures and characterizations
of amenability, and the separation property with respect to compact
quantum subgroups.
Keywords:bicrossed product, locally compact quantum group, noncommutative $L^p$space, positivedefinite function, positivedefinite measure, separation property Categories:20G42, 22D25, 43A35, 46L51, 46L52, 46L89 

16. CJM 2016 (vol 68 pp. 698)
 Skalski, Adam; Sołtan, Piotr

Quantum Families of Invertible Maps and Related Problems
The notion of families of quantum invertible maps (C$^*$algebra
homomorphisms satisfying PodleÅ' condition) is employed to strengthen
and reinterpret several results concerning universal quantum
groups acting on finite quantum spaces. In particular Wang's
quantum automorphism groups are shown to be universal with respect
to quantum families of invertible maps. Further the construction
of the Hopf image of Banica and Bichon is phrased in the purely
analytic language and employed to define the quantum subgroup
generated by a family of quantum subgroups or more generally
a family of quantum invertible maps.
Keywords:quantum families of invertible maps, Hopf image, universal quantum group Categories:46L89, 46L65 

17. CJM 2016 (vol 68 pp. 309)
 Daws, Matthew

Categorical Aspects of Quantum Groups: Multipliers and Intrinsic Groups
We show that the assignment of the (left) completely bounded
multiplier algebra
$M_{cb}^l(L^1(\mathbb G))$ to a locally compact quantum group
$\mathbb G$, and
the assignment of the intrinsic group, form functors between
appropriate
categories. Morphisms of locally compact quantum
groups can be described by Hopf $*$homomorphisms between universal
$C^*$algebras, by bicharacters, or by special sorts of coactions.
We show that the whole
theory of completely bounded multipliers can be lifted to the
universal
$C^*$algebra level, and that then the different pictures of
both multipliers
(reduced, universal, and as centralisers)
and morphisms interact in extremely natural ways. The intrinsic
group of a
quantum group can be realised as a class of multipliers, and
so our techniques
immediately apply. We also show how to think of the intrinsic
group using
the universal $C^*$algebra picture, and then, again, show how
the differing
views on the intrinsic group interact naturally with morphisms.
We show that
the intrinsic group is the ``maximal classical'' quantum subgroup
of a locally
compact quantum group, show that it is even closed in the strong
Vaes sense,
and that the intrinsic group functor is an adjoint to the inclusion
functor
from locally compact groups to quantum groups.
Keywords:locally compact quantum group, morphism, intrinsic group, multiplier, centraliser Categories:20G42, 22D25, 43A22, 43A35, 43A95, 46L52, 46L89, 47L25 

18. CJM 2015 (vol 69 pp. 408)
 Klep, Igor; Špenko, Špela

Free Function Theory Through Matrix Invariants
This paper concerns free function theory. Free maps are free
analogs of analytic functions in several complex variables,
and are defined in terms of freely noncommuting variables.
A function of $g$ noncommuting variables is a function on $g$tuples
of square matrices of all sizes that respects direct sums and
simultaneous conjugation.
Examples of such maps include noncommutative polynomials, noncommutative
rational functions and convergent noncommutative power series.
In sharp contrast to the existing literature in free analysis, this article
investigates free maps \emph{with involution} 
free analogs of real analytic functions.
To
get a grip on these,
techniques and tools from invariant theory are developed and
applied to free analysis. Here is a sample of the results obtained.
A characterization of polynomial free maps via properties of
their finitedimensional slices is presented and then used to
establish power series expansions for analytic free maps about
scalar and nonscalar points; the latter are series of generalized
polynomials for which an invarianttheoretic characterization
is given.
Furthermore,
an inverse and implicit function theorem for free maps with
involution is obtained.
Finally, with a selection of carefully chosen examples
it is shown that
free maps with involution
do not exhibit strong rigidity properties
enjoyed by their involutionfree
counterparts.
Keywords:free algebra, free analysis, invariant theory, polynomial identities, trace identities, concomitants, analytic maps, inverse function theorem, generalized polynomials Categories:16R30, 32A05, 46L52, 15A24, 47A56, 15A24, 46G20 

19. CJM 2015 (vol 67 pp. 1290)
 Charlesworth, Ian; Nelson, Brent; Skoufranis, Paul

On Twofaced Families of Noncommutative Random Variables
We demonstrate that the notions of bifree independence and combinatorialbifree
independence of twofaced families are equivalent using a diagrammatic
view of binoncrossing partitions.
These diagrams produce an operator model on a Fock space suitable
for representing any twofaced family of noncommutative random
variables.
Furthermore, using a Kreweras complement on binoncrossing partitions
we establish the expected formulas for the multiplicative convolution
of a bifree pair of twofaced families.
Keywords:free probability, operator algebras, bifree Category:46L54 

20. CJM 2015 (vol 67 pp. 990)
 Amini, Massoud; Elliott, George A.; Golestani, Nasser

The Category of Bratteli Diagrams
A category structure for Bratteli diagrams is proposed and a
functor from
the category of AF algebras to the category of Bratteli diagrams
is
constructed. Since isomorphism of Bratteli diagrams in this
category coincides
with Bratteli's notion of equivalence, we obtain in particular
a functorial formulation of Bratteli's
classification of AF algebras (and at the same time, of Glimm's
classification of UHF~algebras).
It is shown that the three approaches
to classification of AF~algebras, namely, through Bratteli diagrams,
Ktheory, and
abstract classifying categories, are essentially the same
from a categorical point of view.
Keywords:C$^{*}$algebra, category, functor, AF algebra, dimension group, Bratteli diagram Categories:46L05, 46L35, 46M15 

21. CJM 2015 (vol 67 pp. 481)
 an Huef, Astrid; Archbold, Robert John

The C*algebras of Compact Transformation Groups
We investigate the representation theory of the
crossedproduct $C^*$algebra associated to a compact group $G$
acting on a locally compact space $X$ when the stability subgroups
vary discontinuously.
Our main result applies when $G$ has a principal stability subgroup or
$X$ is locally of finite $G$orbit type. Then the upper multiplicity
of the representation of the crossed product induced from an
irreducible representation $V$ of a stability subgroup is obtained by
restricting $V$ to a certain closed subgroup of the stability subgroup
and taking the maximum of the multiplicities of the irreducible
summands occurring in the restriction of $V$. As a corollary we obtain
that when the trivial subgroup is a principal stability subgroup, the
crossed product is a Fell algebra if and only if every stability
subgroup is abelian. A second corollary is that the $C^*$algebra of
the motion group $\mathbb{R}^n\rtimes \operatorname{SO}(n)$ is a Fell algebra. This uses
the classical branching theorem for the special orthogonal group
$\operatorname{SO}(n)$ with respect to $\operatorname{SO}(n1)$. Since proper transformation
groups are locally induced from the actions of compact groups, we
describe how some of our results can be extended to transformation
groups that are locally proper.
Keywords:compact transformation group, proper action, spectrum of a C*algebra, multiplicity of a representation, crossedproduct C*algebra, continuoustrace C*algebra, Fell algebra Categories:46L05, 46L55 

22. CJM 2015 (vol 67 pp. 870)
 Lin, Huaxin

Minimal Dynamical Systems on Connected Odd Dimensional Spaces
Let $\beta\colon S^{2n+1}\to S^{2n+1}$ be a minimal homeomorphism ($n\ge 1$). We show that
the crossed product $C(S^{2n+1})\rtimes_\beta \mathbb{Z}$ has rational tracial rank at most one.
Let $\Omega$ be a connected compact metric space with finite covering dimension and
with $H^1(\Omega, \mathbb{Z})=\{0\}.$ Suppose that $K_i(C(\Omega))=\mathbb{Z}\oplus G_i,$ where $G_i$ is a finite abelian group, $i=0,1.$
Let $\beta\colon \Omega\to\Omega$ be a minimal homeomorphism. We also show that
$A=C(\Omega)\rtimes_\beta\mathbb{Z}$ has rational tracial rank at most one and is
classifiable.
In particular, this applies to the minimal dynamical systems on
odd dimensional real projective spaces.
This is done by studying minimal homeomorphisms on $X\times \Omega,$ where
$X$ is the Cantor set.
Keywords:minimal dynamical systems Categories:46L35, 46L05 

23. CJM 2014 (vol 67 pp. 404)
 Hua, Jiajie; Lin, Huaxin

Rotation Algebras and the Exel Trace Formula
We found that if $u$ and $v$ are any two unitaries in
a unital $C^*$algebra with $\uvvu\\lt 2$ and $uvu^*v^*$ commutes with
$u$ and $v,$ then the $C^*$subalgebra $A_{u,v}$ generated by $u$ and
$v$ is isomorphic to a quotient of some rotation algebra $A_\theta$
provided that $A_{u,v}$ has a unique tracial state.
We also found that the Exel trace formula holds in any unital
$C^*$algebra.
Let $\theta\in (1/2, 1/2)$ be a real number. We prove the
following:
For any $\epsilon\gt 0,$ there exists $\delta\gt 0$ satisfying the following:
if $u$ and $v$ are two unitaries in any unital simple $C^*$algebra
$A$ with tracial rank zero such that
\[
\uve^{2\pi i\theta}vu\\lt \delta
\text{ and }
{1\over{2\pi i}}\tau(\log(uvu^*v^*))=\theta,
\]
for all tracial state $\tau$ of $A,$ then there exists a pair
of unitaries $\tilde{u}$ and $\tilde{v}$ in $A$
such that
\[
\tilde{u}\tilde{v}=e^{2\pi i\theta} \tilde{v}\tilde{u},\,\,
\u\tilde{u}\\lt \epsilon
\text{ and }
\v\tilde{v}\\lt \epsilon.
\]
Keywords:rotation algebras, Exel trace formula Category:46L05 

24. CJM 2013 (vol 65 pp. 1287)
 Reihani, Kamran

$K$theory of Furstenberg Transformation Group $C^*$algebras
The paper studies the $K$theoretic invariants of the crossed product
$C^{*}$algebras associated with an important family of homeomorphisms
of the tori $\mathbb{T}^{n}$ called Furstenberg transformations.
Using the PimsnerVoiculescu theorem, we prove that given $n$, the
$K$groups of those crossed products, whose corresponding $n\times n$
integer matrices are unipotent of maximal degree, always have the same
rank $a_{n}$. We show using the theory developed here that a claim
made in the literature about the torsion subgroups of these $K$groups
is false. Using the representation theory of the simple Lie algebra
$\frak{sl}(2,\mathbb{C})$, we show that, remarkably, $a_{n}$ has a
combinatorial significance. For example, every $a_{2n+1}$ is just the
number of ways that $0$ can be represented as a sum of integers
between $n$ and $n$ (with no repetitions). By adapting an argument
of van Lint (in which he answered a question of ErdÅs), a simple,
explicit formula for the asymptotic behavior of the sequence
$\{a_{n}\}$ is given. Finally, we describe the order structure of the
$K_{0}$groups of an important class of Furstenberg crossed products,
obtaining their complete Elliott invariant using classification
results of H. Lin and N. C. Phillips.
Keywords:$K$theory, transformation group $C^*$algebra, Furstenberg transformation, Anzai transformation, minimal homeomorphism, positive cone, minimal homeomorphism Categories:19K14, 19K99, 46L35, 46L80, , 05A15, 05A16, 05A17, 15A36, 17B10, 17B20, 37B05, 54H20 

25. 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 
