1. 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 

2. 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 

3. 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 

4. 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 

5. CJM 2005 (vol 57 pp. 983)
6. CJM 2003 (vol 55 pp. 1302)
7. CJM 1999 (vol 51 pp. 745)
 Echterhoff, Siegfried; Quigg, John

Induced Coactions of Discrete Groups on $C^*$Algebras
Using the close relationship between coactions of discrete groups and
Fell bundles, we introduce a procedure for inducing a $C^*$coaction
$\delta\colon D\to D\otimes C^*(G/N)$ of a quotient group $G/N$ of a
discrete group $G$ to a $C^*$coaction $\Ind\delta\colon\Ind D\to \Ind
D\otimes C^*(G)$ of $G$. We show that induced coactions behave in many
respects similarly to induced actions. In particular, as an analogue of
the well known imprimitivity theorem for induced actions we prove that
the crossed products $\Ind D\times_{\Ind\delta}G$ and $D\times_{\delta}G/N$
are always Morita equivalent. We also obtain nonabelian analogues of a
theorem of Olesen and Pedersen which show that there is a duality between
induced coactions and twisted actions in the sense of Green. We further
investigate amenability of Fell bundles corresponding to induced coactions.
Category:46L55 
