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
 Glöckner, Helge

Completeness of infinitedimensional Lie groups in their left uniformity
We prove completeness for the main examples
of infinitedimensional Lie groups and some related topological
groups.
Consider a sequence
$G_1\subseteq G_2\subseteq\cdots$ of topological groups~$G_n$
such that~$G_n$ is a subgroup of $G_{n+1}$ and the latter induces
the given topology on~$G_n$,
for each $n\in\mathbb{N}$.
Let $G$ be the direct limit of the sequence in the category of
topological groups.
We show that $G$ induces the given topology on each~$G_n$ whenever
$\bigcup_{n\in \mathbb{N}}V_1V_2\cdots V_n$ is an identity neighbourhood
in~$G$
for all identity neighbourhoods $V_n\subseteq G_n$. If, moreover,
each $G_n$ is complete, then~$G$ is complete.
We also show that the weak direct product $\bigoplus_{j\in J}G_j$
is complete for
each family $(G_j)_{j\in J}$ of complete Lie groups~$G_j$.
As a consequence, every strict direct limit $G=\bigcup_{n\in
\mathbb{N}}G_n$ of finitedimensional
Lie groups is complete, as well as the diffeomorphism group
$\operatorname{Diff}_c(M)$
of a paracompact finitedimensional smooth manifold~$M$
and the test function group $C^k_c(M,H)$, for each $k\in\mathbb{N}_0\cup\{\infty\}$
and complete Lie group~$H$
modelled on a complete locally convex space.
Keywords:infinitedimensional Lie group, left uniform structure, completeness Categories:22E65, 22A05, 22E67, 46A13, 46M40, 58D05 

3. 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 

4. CJM Online first
 Elazar, Boaz; Shaviv, Ary

Schwartz functions on real algebraic varieties
We define Schwartz functions, tempered functions and tempered
distributions on (possibly singular) real algebraic varieties.
We prove that all classical properties of these spaces, defined
previously on affine spaces and on Nash manifolds, also hold
in the case of affine real algebraic varieties, and give partial
results for the nonaffine case.
Keywords:real algebraic geometry, Schwartz function, tempered distribution Categories:14P99, 14P05, 22E45, 46A11, 46F05 

5. CJM Online first
 Handelman, David

Nearly approximate transitivity (AT) for circulant matrices
By previous work of Giordano and the author, ergodic
actions of $\mathbf Z$ (and other discrete groups) are completely classified
measuretheoretically by their dimension space, a construction
analogous to the dimension group used in C*algebras and topological
dynamics. Here we investigate how far from AT (approximately
transitive) can actions be which derive from circulant (and related)
matrices. It turns out not very: although nonAT actions can
arise from this method of construction, under very modest additional
conditions, ATness arises; in addition, if we drop the positivity
requirement in the isomorphism of dimension spaces, then all
these ergodic actions satisfy an analogue of AT. Many examples
are provided.
Keywords:approximately transitive, ergodic transformation, circulant matrix, hemicirculant matrix, dimension space, matrixvalued random walk Categories:37A05, 06F25, 28D05, 46B40, 60G50 

6. 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 

7. 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 

8. 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 

9. 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 

10. 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 

11. 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 

12. CJM Online first
 Dantas, Sheldon; García, Domingo; Maestre, Manuel; Martín, Miguel

The BishopPhelpsBollobÃ¡s property for compact operators
We study the BishopPhelpsBollobÃ¡s property (BPBp for short)
for compact operators. We present some abstract techniques which
allows to carry the BPBp for compact operators from sequence
spaces to function spaces. As main applications, we prove the
following results. Let $X$, $Y$ be Banach spaces. If $(c_0,Y)$
has the BPBp for compact operators, then so do $(C_0(L),Y)$ for
every locally compact Hausdorff topological space $L$ and $(X,Y)$
whenever $X^*$ is isometrically isomorphic to $\ell_1$.
If $X^*$ has the RadonNikodÃ½m property and $(\ell_1(X),Y)$
has the BPBp for compact operators, then so does $(L_1(\mu,X),Y)$
for every positive measure $\mu$; as a consequence, $(L_1(\mu,X),Y)$
has the the BPBp for compact operators when $X$ and $Y$ are finitedimensional
or $Y$ is a Hilbert space and $X=c_0$ or $X=L_p(\nu)$ for any
positive measure $\nu$ and $1\lt p\lt \infty$.
For $1\leq p \lt \infty$, if $(X,\ell_p(Y))$ has the BPBp for compact
operators, then so does $(X,L_p(\mu,Y))$ for every positive measure
$\mu$ such that $L_1(\mu)$ is infinitedimensional. If $(X,Y)$
has the BPBp for compact operators, then so do $(X,L_\infty(\mu,Y))$
for every $\sigma$finite positive measure $\mu$ and $(X,C(K,Y))$
for every compact Hausdorff topological space $K$.
Keywords:BishopPhelps theorem, BishopPhelpsBollobÃ¡s property, norm attaining operator, compact operator Categories:46B04, 46B20, 46B28, 46B25, 46E40 

13. CJM Online first
 Bao, Guanlong; Göğüş, Nihat Gökhan; Pouliasis, Stamatis

On Dirichlet spaces with a class of superharmonic weights
In this paper, we investigate Dirichlet spaces $\mathcal{D}_\mu$ with
superharmonic weights induced by positive Borel measures $\mu$
on
the open unit disk. We establish the AlexanderTaylorUllman
inequality for $\mathcal{D}_\mu$ spaces and we characterize the cases where
equality occurs.
We define a class of weighted Hardy spaces $H_{\mu}^{2}$ via
the balayage of the measure $\mu$.
We show that $\mathcal{D}_\mu$
is equal to $H_{\mu}^{2}$ if and only if $\mu$ is a
Carleson measure for $\mathcal{D}_\mu$. As an application, we obtain the
reproducing kernel of $\mathcal{D}_\mu$ when $\mu$ is an infinite
sum of point mass measures. We consider the boundary
behavior and innerouter factorization of functions in $\mathcal{D}_\mu$.
We also characterize the boundedness and
compactness of composition operators on $\mathcal{D}_\mu$.
Keywords:Dirichlet space, Hardy space, superharmonic weight Categories:30H10, 31C25, 46E15 

14. CJM 2017 (vol 69 pp. 1312)
 Fricain, Emmanuel; Rupam, Rishika

On Asymptotically Orthonormal Sequences
An asymptotically orthonormal sequence is a sequence which is
"nearly" orthonormal in the sense that it satisfies the Parseval
equality up to two constants close to one. In this paper, we
explore such sequences formed by normalized reproducing kernels
for model spaces and de BrangesRovnyak spaces.
Keywords:function space, de BrangesRovnyak and model space, reproducing kernel, asymptotically orthonormal sequence Categories:30J05, 30H10, 46E22 

15. CJM Online first
 Giannopoulos, Apostolos; Koldobsky, Alexander; Valettas, Petros

Inequalities for the surface area of projections of convex bodies
We provide general inequalities that compare the surface area
$S(K)$ of a convex body $K$ in ${\mathbb R}^n$
to the minimal, average or maximal surface area of its hyperplane
or lower dimensional projections. We discuss the
same questions for all the quermassintegrals of $K$. We examine
separately the dependence of the constants
on the dimension in the case where $K$ is in some of the classical
positions or $K$ is a projection body.
Our results are in the spirit of the hyperplane problem, with
sections replaced by projections and volume by
surface area.
Keywords:surface area, convex body, projection Categories:52A20, 46B05 

16. 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 

17. 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 

18. CJM 2016 (vol 69 pp. 650)
 Oikhberg, Timur; Tradacete, Pedro

Almost Disjointness Preservers
We study the stability of disjointness preservers on Banach lattices.
In many cases, we prove that an "almost disjointness preserving"
operator is well approximable by a disjointness preserving one.
However, this approximation is not always possible, as our
examples show.
Keywords:Banach lattice, disjointness preserving Categories:47B38, 46B42 

19. 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 

20. 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 

21. 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 

22. 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 

23. CJM 2016 (vol 69 pp. 54)
 Hartz, Michael

On the Isomorphism Problem for Multiplier Algebras of NevanlinnaPick Spaces
We continue the investigation of the isomorphism problem for
multiplier algebras of reproducing kernel
Hilbert spaces with the complete NevanlinnaPick property.
In contrast to previous work in this area,
we do not study these spaces by identifying them with restrictions
of a universal space, namely the DruryArveson space.
Instead, we work directly with the Hilbert spaces and their
reproducing kernels. In particular,
we show that two multiplier algebras of NevanlinnaPick spaces
on the same set are equal if and only if the Hilbert
spaces are equal. Most of the article is devoted to the study
of a special class of
complete NevanlinnaPick spaces on homogeneous varieties. We
provide a complete
answer to the question of when two multiplier algebras of spaces
of this type
are algebraically or isometrically isomorphic. This generalizes
results of Davidson, Ramsey, Shalit,
and the author.
Keywords:nonselfadjoint operator algebras, reproducing kernel Hilbert spaces, multiplier algebra, NevanlinnaPick kernels, isomorphism problem Categories:47L30, 46E22, 47A13 

24. 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 

25. 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 
