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1. CJM Online first

Luo, Caihua
 Spherical fundamental lemma for metaplectic groups In this paper, we prove the spherical fundamental lemma for metaplectic group $Mp_{2n}$ based on the formalism of endoscopy theory by J.Adams, D.Renard and Wen-Wei Li. Keywords:metaplectic group, endoscopic group, elliptic stable trace formula, fundamental lemmaCategory:22E35

2. CJM 2016 (vol 68 pp. 1382)

Zydor, Michał
 La Variante infinitÃ©simale de la formule des traces de Jacquet-Rallis pour les groupes unitaires We establish an infinitesimal version of the Jacquet-Rallis 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 semi-simple classes $\mathfrak{o}$, $J_{\mathfrak{o}}$ is a relative orbital integral of Jacquet-Rallis. For classes $\mathfrak{o}$ called relatively regular semi-simple, we express $J_{\mathfrak{o}}$ in terms of relative orbital integrals regularised by means of zÃªta functions. Keywords:formule des traces relativeCategories:11F70, 11F72

3. CJM 2016 (vol 68 pp. 908)

Sugiyama, Shingo; Tsuzuki, Masao
 Existence of Hilbert Cusp Forms with Non-vanishing $L$-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 non-vanishing 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 $L$-functionsCategories:11F67, 11F72

4. 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 finite-dimensional slices is presented and then used to establish power series expansions for analytic free maps about scalar and non-scalar points; the latter are series of generalized polynomials for which an invariant-theoretic 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 involution-free counterparts. Keywords:free algebra, free analysis, invariant theory, polynomial identities, trace identities, concomitants, analytic maps, inverse function theorem, generalized polynomialsCategories:16R30, 32A05, 46L52, 15A24, 47A56, 15A24, 46G20

5. 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 crossed-product $C^*$-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 $\operatorname{SO}(n-1)$. 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, crossed-product C*-algebra, continuous-trace C*-algebra, Fell algebraCategories:46L05, 46L55

6. 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 $\|uv-vu\|\lt 2$ 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 $\|uv-e^{2\pi i\theta}vu\|\lt \delta \text{ and } {1\over{2\pi i}}\tau(\log(uvu^*v^*))=\theta,$ 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 formulaCategory:46L05

7. CJM 2012 (vol 64 pp. 497)

Li, Wen-Wei

8. 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 self-adjoint elements $H_0$ and $H$ in the algebra and relate it to the concept of the de la Harpe--Skandalis 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 Birman--Solomyak formula for a general non-linear perturbation, (ii) a universality of a spectral averaging, and (iii) a generalization of the Dixmier--Fuglede--Kadison differentiation formula. Keywords:perturbation determinant, trace formulae, von Neumann algebrasCategories:47A55, 47C15, 47A53

9. 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, traceCategories:11M41, 11F30, , 11F55, 11H06, 11R47

10. 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 pseudo-coefficient 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, pseudo-coefficientCategory:22E50

11. 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~Neumann-Schatten trace ideals. Banach spaces that are $q$-uniformly $\PL$-convex in the sense of Davis, Garling and Tomczak-Jaegermann 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 idealsCategories:46B20, 46L52

12. 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 $c^{k-1}$. 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, multitraceCategory:08B26

13. CJM 2001 (vol 53 pp. 631)

Walters, Samuel G.
 K-Theory of Non-Commutative 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 nine-dimensional 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, K-theory, automorphisms, rotation algebras, unbounded traces, Chern charactersCategories:46L80, 46L40, 19K14

14. CJM 2000 (vol 52 pp. 633)

Walters, Samuel G.
 Chern Characters of Fourier Modules Let $A_\theta$ denote the rotation algebra---the 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 modules---and more importantly, the Fourier module---are 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 non-commutative 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, $K$-groupsCategories:46L80, 46L40

15. CJM 1999 (vol 51 pp. 952)

Deitmar, Anton; Hoffmann, Werner
 On Limit Multiplicities for Spaces of Automorphic Forms Let $\Gamma$ be a rank-one 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 square-integrable $\Gamma$-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 DeGeorge-Wallach and Delorme. Keywords:limit multiplicities, automorphic forms, noncompact quotients, Selberg trace formula, functional calculusCategories:11F72, 22E30, 22E40, 43A85, 58G25
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