1. CJM 2016 (vol 68 pp. 1382)
 Zydor, Michał

La Variante infinitÃ©simale de la formule des traces de JacquetRallis pour les groupes unitaires
We establish an infinitesimal version of the
JacquetRallis trace formula for unitary groups.
Our formula is obtained by integrating a
truncated kernel Ã la Arthur.
It has a geometric side which is a
sum of distributions $J_{\mathfrak{o}}$ indexed by classes of
elements
of the Lie algebra of $U(n+1)$ stable by $U(n)$conjugation
as well as the "spectral side"
consisting of the Fourier transforms
of the aforementioned distributions.
We prove that the distributions $J_{\mathfrak{o}}$
are invariant and depend only on the choice of
the Haar measure on $U(n)(\mathbb{A})$.
For regular semisimple classes $\mathfrak{o}$, $J_{\mathfrak{o}}$
is
a relative orbital integral of JacquetRallis.
For classes $\mathfrak{o}$ called relatively regular semisimple,
we express $J_{\mathfrak{o}}$
in terms of relative orbital integrals regularised by means of
zÃªta functions.
Keywords:formule des traces relative Categories:11F70, 11F72 

2. CJM 2016 (vol 68 pp. 908)
 Sugiyama, Shingo; Tsuzuki, Masao

Existence of Hilbert Cusp Forms with Nonvanishing $L$values
We develop a derivative version of the relative trace formula
on $\operatorname{PGL}(2)$ studied in our previous work,
and derive an asymptotic formula of an average of central values
(derivatives)
of automorphic $L$functions for Hilbert cusp forms.
As an application, we prove the existence of Hilbert cusp forms
with nonvanishing central values (derivatives)
such that the absolute degrees of their Hecke fields are arbitrarily
large.
Keywords:automorphic representations, relative trace formulas, central $L$values, derivatives of $L$functions Categories:11F67, 11F72 

3. 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 

4. 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 

5. 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 

6. CJM 2012 (vol 64 pp. 497)
 Li, WenWei

Le lemme fondamental pondÃ©rÃ© pour le groupe mÃ©taplectique
Dans cet article, on Ã©nonce une variante du lemme fondamental
pondÃ©rÃ© d'Arthur pour le groupe mÃ©taplectique de Weil, qui sera un
ingrÃ©dient indispensable de la stabilisation de la formule des
traces. Pour un corps de caractÃ©ristique rÃ©siduelle suffisamment
grande, on en donne une dÃ©monstration Ã l'aide de la mÃ©thode de
descente, qui est conditionnelle: on admet le lemme fondamental
pondÃ©rÃ© non standard sur les algÃ¨bres de Lie. Vu les travaux de
Chaudouard et Laumon, on s'attend Ã ce que cette condition soit
ultÃ©rieurement vÃ©rifiÃ©e.
Keywords:fundamental lemma, metaplectic group, endoscopy, trace formula Categories:11F70, 11F27, 22E50 

7. CJM 2009 (vol 62 pp. 133)
 Makarov, Konstantin A.; Skripka, Anna

Some Applications of the Perturbation Determinant in Finite von Neumann Algebras
In the finite von Neumann algebra setting, we introduce the concept
of a perturbation determinant associated with a pair of selfadjoint
elements $H_0$ and $H$ in the algebra and relate it to the concept of
the de la HarpeSkandalis homotopy invariant determinant associated
with piecewise $C^1$paths of operators joining $H_0$ and $H$. We
obtain an analog of Krein's formula that relates the perturbation
determinant and the spectral shift function and, based on this
relation, we derive subsequently (i) the BirmanSolomyak formula for
a general nonlinear perturbation, (ii) a universality of a spectral
averaging, and (iii) a generalization of the
DixmierFugledeKadison differentiation formula.
Keywords:perturbation determinant, trace formulae, von Neumann algebras Categories:47A55, 47C15, 47A53 

8. CJM 2007 (vol 59 pp. 673)
 Ash, Avner; Friedberg, Solomon

Hecke $L$Functions and the Distribution of Totally Positive Integers
Let $K$ be a totally real number field of degree $n$. We show that
the number of totally positive integers
(or more generally the number of totally positive elements of a given fractional ideal)
of given trace is evenly distributed around its expected value, which is
obtained from geometric considerations.
This result depends on unfolding an integral over
a compact torus.
Keywords:Eisenstein series, toroidal integral, Fourier series, Hecke $L$function, totally positive integer, trace Categories:11M41, 11F30, , 11F55, 11H06, 11R47 

9. CJM 2006 (vol 58 pp. 1229)
 Henniart, Guy; Lemaire, Bertrand

IntÃ©grales orbitales tordues sur $\GL(n,F)$ et corps locaux proches\,: applications
Soient $F$ un corps
commutatif localement compact non archim\'edien, $G=\GL
(n,F)$ pour un entier $n\geq 2$, et $\kappa$ un caract\`ere de
$F^\times$ trivial sur $(F^\times)^n$. On prouve une formule pour
les $\kappa$int\'egrales orbitales r\'eguli\`eres sur $G$
permettant, si $F$ est de caract\'eristique $>0$, de les relever
\`a la caract\'eristique nulle. On en d\'eduit deux r\'esultats
nouveaux en caract\'eristique $>0$\,: le ``lemme fondamental'' pour
l'induction automorphe, et une version simple de la formule des
traces tordue locale d'Arthur reliant $\kappa$int\'egrales
orbitales elliptiques et caract\`eres $\kappa$tordus. Cette
formule donne en particulier, pour une s\'erie
$\kappa$discr\`ete de $G$, les $\kappa$int\'egrales orbitales
elliptiques d'un pseudocoefficient comme valeurs du caract\`ere
$\kappa$tordu.
Keywords:corps local, reprÃ©sentation lisse, intÃ©grale orbitale tordue, induction automorphe, lemme fondamental, formule des traces locale, pseudocoefficient Category:22E50 

10. CJM 2004 (vol 56 pp. 225)
 Blower, Gordon; Ransford, Thomas

Complex Uniform Convexity and Riesz Measure
The norm on a Banach space gives rise to a subharmonic function on the
complex plane for which the distributional Laplacian gives a Riesz measure.
This measure is calculated explicitly here for Lebesgue $L^p$ spaces and the
von~NeumannSchatten trace ideals. Banach spaces that are $q$uniformly
$\PL$convex in the sense of Davis, Garling and TomczakJaegermann are
characterized in terms of the mass distribution of this measure. This gives
a new proof that the trace ideals $c^p$ are $2$uniformly $\PL$convex for
$1\leq p\leq 2$.
Keywords:subharmonic functions, Banach spaces, Schatten trace ideals Categories:46B20, 46L52 

11. CJM 2002 (vol 54 pp. 736)
 Kearnes, K. A.; Kiss, E. W.; Szendrei, Á.; Willard, R. D.

Chief Factor Sizes in Finitely Generated Varieties
Let $\mathbf{A}$ be a $k$element algebra whose chief factor size is
$c$. We show that if $\mathbf{B}$ is in the variety generated by
$\mathbf{A}$, then any abelian chief factor of $\mathbf{B}$ that is
not strongly abelian has size at most $c^{k1}$. This solves
Problem~5 of {\it The Structure of Finite Algebras}, by D.~Hobby and
R.~McKenzie. We refine this bound to $c$ in the situation where the
variety generated by $\mathbf{A}$ omits type $\mathbf{1}$. As a
generalization, we bound the size of multitraces of types~$\mathbf{1}$,
$\mathbf{2}$, and $\mathbf{3}$ by extending coordinatization
theory. Finally, we exhibit some examples of bad behavior, even in
varieties satisfying a congruence identity.
Keywords:tame congruence theory, chief factor, multitrace Category:08B26 

12. CJM 2001 (vol 53 pp. 631)
 Walters, Samuel G.

KTheory of NonCommutative Spheres Arising from the Fourier Automorphism
For a dense $G_\delta$ set of real parameters $\theta$ in $[0,1]$
(containing the rationals) it is shown that the group $K_0 (A_\theta
\rtimes_\sigma \mathbb{Z}_4)$ is isomorphic to $\mathbb{Z}^9$, where
$A_\theta$ is the rotation C*algebra generated by unitaries $U$, $V$
satisfying $VU = e^{2\pi i\theta} UV$ and $\sigma$ is the Fourier
automorphism of $A_\theta$ defined by $\sigma(U) = V$, $\sigma(V) =
U^{1}$. More precisely, an explicit basis for $K_0$ consisting of
nine canonical modules is given. (A slight generalization of this
result is also obtained for certain separable continuous fields of
unital C*algebras over $[0,1]$.) The Connes Chern character $\ch
\colon K_0 (A_\theta \rtimes_\sigma \mathbb{Z}_4) \to H^{\ev} (A_\theta
\rtimes_\sigma \mathbb{Z}_4)^*$ is shown to be injective for a dense
$G_\delta$ set of parameters $\theta$. The main computational tool in
this paper is a group homomorphism $\vtr \colon K_0 (A_\theta
\rtimes_\sigma \mathbb{Z}_4) \to \mathbb{R}^8 \times \mathbb{Z}$
obtained from the Connes Chern character by restricting the
functionals in its codomain to a certain ninedimensional subspace of
$H^{\ev} (A_\theta \rtimes_\sigma \mathbb{Z}_4)$. The range of $\vtr$
is fully determined for each $\theta$. (We conjecture that this
subspace is all of $H^{\ev}$.)
Keywords:C*algebras, Ktheory, automorphisms, rotation algebras, unbounded traces, Chern characters Categories:46L80, 46L40, 19K14 

13. CJM 2000 (vol 52 pp. 633)
 Walters, Samuel G.

Chern Characters of Fourier Modules
Let $A_\theta$ denote the rotation algebrathe universal $C^\ast$algebra
generated by unitaries $U,V$ satisfying $VU=e^{2\pi i\theta}UV$, where
$\theta$ is a fixed real number. Let $\sigma$ denote the Fourier
automorphism of $A_\theta$ defined by $U\mapsto V$, $V\mapsto U^{1}$,
and let $B_\theta = A_\theta \rtimes_\sigma \mathbb{Z}/4\mathbb{Z}$ denote
the associated $C^\ast$crossed product. It is shown that there is a
canonical inclusion $\mathbb{Z}^9 \hookrightarrow K_0(B_\theta)$ for each
$\theta$ given by nine canonical modules. The unbounded trace functionals
of $B_\theta$ (yielding the Chern characters here) are calculated to obtain
the cyclic cohomology group of order zero $\HC^0(B_\theta)$ when
$\theta$ is irrational. The Chern characters of the nine modulesand more
importantly, the Fourier moduleare computed and shown to involve techniques
from the theory of Jacobi's theta functions. Also derived are explicit
equations connecting unbounded traces across strong Morita equivalence, which
turn out to be noncommutative extensions of certain theta function equations.
These results provide the basis for showing that for a dense $G_\delta$ set
of values of $\theta$ one has $K_0(B_\theta)\cong\mathbb{Z}^9$ and is
generated by the nine classes constructed here.
Keywords:$C^\ast$algebras, unbounded traces, Chern characters, irrational rotation algebras, $K$groups Categories:46L80, 46L40 

14. CJM 1999 (vol 51 pp. 952)
 Deitmar, Anton; Hoffmann, Werner

On Limit Multiplicities for Spaces of Automorphic Forms
Let $\Gamma$ be a rankone arithmetic subgroup of a
semisimple Lie group~$G$. For fixed $K$Type, the spectral
side of the Selberg trace formula defines a distribution
on the space of infinitesimal characters of~$G$, whose
discrete part encodes the dimensions of the spaces of
squareintegrable $\Gamma$automorphic forms. It is shown
that this distribution converges to the Plancherel measure
of $G$ when $\Ga$ shrinks to the trivial group in a certain
restricted way. The analogous assertion for cocompact
lattices $\Gamma$ follows from results of DeGeorgeWallach
and Delorme.
Keywords:limit multiplicities, automorphic forms, noncompact quotients, Selberg trace formula, functional calculus Categories:11F72, 22E30, 22E40, 43A85, 58G25 
