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26. CJM 2006 (vol 58 pp. 691)

Bendikov, A.; Saloff-Coste, L.
 Hypoelliptic Bi-Invariant Laplacians on Infinite Dimensional Compact Groups On a compact connected group $G$, consider the infinitesimal generator $-L$ of a central symmetric Gaussian convolution semigroup $(\mu_t)_{t>0}$. Using appropriate notions of distribution and smooth function spaces, we prove that $L$ is hypoelliptic if and only if $(\mu_t)_{t>0}$ is absolutely continuous with respect to Haar measure and admits a continuous density $x\mapsto \mu_t(x)$, $t>0$, such that $\lim_{t\ra 0} t\log \mu_t(e)=0$. In particular, this condition holds if and only if any Borel measure $u$ which is solution of $Lu=0$ in an open set $\Omega$ can be represented by a continuous function in $\Omega$. Examples are discussed. Categories:60B15, 43A77, 35H10, 46F25, 60J45, 60J60

27. CJM 2005 (vol 57 pp. 1193)

Dungey, Nick
 Some Conditions for Decay of Convolution Powers and Heat Kernels on Groups Let $K$ be a function on a unimodular locally compact group $G$, and denote by $K_n = K*K* \cdots * K$ the $n$-th convolution power of $K$. Assuming that $K$ satisfies certain operator estimates in $L^2(G)$, we give estimates of the norms $\|K_n\|_2$ and $\|K_n\|_\infty$ for large $n$. In contrast to previous methods for estimating $\|K_n\|_\infty$, we do not need to assume that the function $K$ is a probability density or non-negative. Our results also adapt for continuous time semigroups on $G$. Various applications are given, for example, to estimates of the behaviour of heat kernels on Lie groups. Categories:22E30, 35B40, 43A99

28. CJM 2005 (vol 57 pp. 598)

Kornelson, Keri A.
 Local Solvability of Laplacian Difference Operators Arising from the Discrete Heisenberg Group Differential operators $D_x$, $D_y$, and $D_z$ are formed using the action of the $3$-dimensional discrete Heisenberg group $G$ on a set $S$, and the operators will act on functions on $S$. The Laplacian operator $L=D_x^2 + D_y^2 + D_z^2$ is a difference operator with variable differences which can be associated to a unitary representation of $G$ on the Hilbert space $L^2(S)$. Using techniques from harmonic analysis and representation theory, we show that the Laplacian operator is locally solvable. Keywords:discrete Heisenberg group,, unitary representation,, local solvability,, difference operatorCategories:43A85, 22D10, 39A70

29. CJM 2005 (vol 57 pp. 99)

Fegan, H. D.; Steer, B.
 Second Order Operators on a Compact Lie Group We describe the structure of the space of second order elliptic differential operators on a homogenous bundle over a compact Lie group. Subject to a technical condition, these operators are homotopic to the Laplacian. The technical condition is further investigated, with examples given where it holds and others where it does not. Since many spectral invariants are also homotopy invariants, these results provide information about the invariants of these operators. Categories:58J70, 43A77

30. CJM 2005 (vol 57 pp. 17)

Bédos, Erik; Conti, Roberto; Tuset, Lars
 On Amenability and Co-Amenability of Algebraic Quantum Groups and Their Corepresentations We introduce and study several notions of amenability for unitary corepresentations and $*$-representations of algebraic quantum groups, which may be used to characterize amenability and co-amenability for such quantum groups. As a background for this study, we investigate the associated tensor C$^{*}$-categories. Keywords:quantum group, amenabilityCategories:46L05, 46L65, 22D10, 22D25, 43A07, 43A65, 58B32

31. CJM 2004 (vol 56 pp. 1259)

Paterson, Alan L. T.
 The Fourier Algebra for Locally Compact Groupoids We introduce and investigate using Hilbert modules the properties of the {\em Fourier algebra} $A(G)$ for a locally compact groupoid $G$. We establish a duality theorem for such groupoids in terms of multiplicative module maps. This includes as a special case the classical duality theorem for locally compact groups proved by P. Eymard. Keywords:Fourier algebra, locally compact groupoids, Hilbert modules,, positive definite functions, completely bounded mapsCategory:43A32

32. CJM 2004 (vol 56 pp. 344)

Miao, Tianxuan
 Predual of the Multiplier Algebra of $A_p(G)$ and Amenability For a locally compact group $G$ and $1 Keywords:Locally compact groups, amenable groups, multiplier algebra, Herz algebraCategory:43A07 33. CJM 2004 (vol 56 pp. 431) Rosenblatt, Joseph; Taylor, Michael  Group Actions and Singular Martingales II, The Recognition Problem We continue our investigation in [RST] of a martingale formed by picking a measurable set$A$in a compact group$G$, taking random rotates of$A$, and considering measures of the resulting intersections, suitably normalized. Here we concentrate on the inverse problem of recognizing$A$from a small amount of data from this martingale. This leads to problems in harmonic analysis on$G$, including an analysis of integrals of products of Gegenbauer polynomials. Categories:43A77, 60B15, 60G42, 42C10 34. CJM 2003 (vol 55 pp. 1134) Casarino, Valentina  Norms of Complex Harmonic Projection Operators In this paper we estimate the$(L^p-L^2)$-norm of the complex harmonic projectors$\pi_{\ell\ell'}$,$1\le p\le 2$, uniformly with respect to the indexes$\ell,\ell'$. We provide sharp estimates both for the projectors$\pi_{\ell\ell'}$, when$\ell,\ell'$belong to a proper angular sector in$\mathbb{N} \times \mathbb{N}$, and for the projectors$\pi_{\ell 0}$and$\pi_{0 \ell}$. The proof is based on an extension of a complex interpolation argument by C.~Sogge. In the appendix, we prove in a direct way the uniform boundedness of a particular zonal kernel in the$L^1$norm on the unit sphere of$\mathbb{R}^{2n}$. Categories:43A85, 33C55, 42B15 35. CJM 2003 (vol 55 pp. 1000) Graczyk, P.; Sawyer, P.  Some Convexity Results for the Cartan Decomposition In this paper, we consider the set$\mathcal{S} = a(e^X K e^Y)$where$a(g)$is the abelian part in the Cartan decomposition of$g$. This is exactly the support of the measure intervening in the product formula for the spherical functions on symmetric spaces of noncompact type. We give a simple description of that support in the case of$\SL(3,\mathbf{F})$where$\mathbf{F} = \mathbf{R}$,$\mathbf{C}$or$\mathbf{H}$. In particular, we show that$\mathcal{S}$is convex. We also give an application of our result to the description of singular values of a product of two arbitrary matrices with prescribed singular values. Keywords:convexity theorems, Cartan decomposition, spherical functions, product formula, semisimple Lie groups, singular valuesCategories:43A90, 53C35, 15A18 36. CJM 2002 (vol 54 pp. 1280) Skrzypczak, Leszek  Besov Spaces and Hausdorff Dimension For Some Carnot-CarathÃ©odory Metric Spaces We regard a system of left invariant vector fields$\mathcal{X}=\{X_1,\dots,X_k\}$satisfying the H\"ormander condition and the related Carnot-Carath\'eodory metric on a unimodular Lie group$G$. We define Besov spaces corresponding to the sub-Laplacian$\Delta=\sum X_i^2$both with positive and negative smoothness. The atomic decomposition of the spaces is given. In consequence we get the distributional characterization of the Hausdorff dimension of Borel subsets with the Haar measure zero. Keywords:Besov spaces, sub-elliptic operators, Carnot-CarathÃ©odory metric, Hausdorff dimensionCategories:46E35, 43A15, 28A78 37. CJM 2002 (vol 54 pp. 1100) Wood, Peter J.  The Operator Biprojectivity of the Fourier Algebra In this paper, we investigate projectivity in the category of operator spaces. In particular, we show that the Fourier algebra of a locally compact group$G$is operator biprojective if and only if$G$is discrete. Keywords:locally compact group, Fourier algebra, operator space, projectiveCategories:13D03, 18G25, 43A95, 46L07, 22D99 38. CJM 2002 (vol 54 pp. 634) Weber, Eric  Frames and Single Wavelets for Unitary Groups We consider a unitary representation of a discrete countable abelian group on a separable Hilbert space which is associated to a cyclic generalized frame multiresolution analysis. We extend Robertson's theorem to apply to frames generated by the action of the group. Within this setup we use Stone's theorem and the theory of projection valued measures to analyze wandering frame collections. This yields a functional analytic method of constructing a wavelet from a generalized frame multi\-resolution analysis in terms of the frame scaling vectors. We then explicitly apply our results to the action of the integers given by translations on$L^2({\mathbb R})$. Keywords:wavelet, multiresolution analysis, unitary group representation, frameCategories:42C40, 43A25, 42C15, 46N99 39. CJM 2002 (vol 54 pp. 303) Ghahramani, Fereidoun; Grabiner, Sandy  Convergence Factors and Compactness in Weighted Convolution Algebras We study convergence in weighted convolution algebras$L^1(\omega)$on$R^+$, with the weights chosen such that the corresponding weighted space$M(\omega)$of measures is also a Banach algebra and is the dual space of a natural related space of continuous functions. We determine convergence factor$\eta$for which weak$^\ast$-convergence of$\{\lambda_n\}$to$\lambda$in$M(\omega)$implies norm convergence of$\lambda_n \ast f$to$\lambda \ast f$in$L^1 (\omega\eta)$. We find necessary and sufficent conditions which depend on$\omega$and$f$and also find necessary and sufficent conditions for$\eta$to be a convergence factor for all$L^1(\omega)$and all$f$in$L^1(\omega)$. We also give some applications to the structure of weighted convolution algebras. As a preliminary result we observe that$\eta$is a convergence factor for$\omega$and$f$if and only if convolution by$f$is a compact operator from$M(\omega)$(or$L^1(\omega)$) to$L^1 (\omega\eta)$. Categories:43A10, 43A15, 46J45, 46J99 40. CJM 2001 (vol 53 pp. 944) Ludwig, J.; Molitor-Braun, C.  ReprÃ©sentations irrÃ©ductibles bornÃ©es des groupes de Lie exponentiels Let$G$be a solvable exponential Lie group. We characterize all the continuous topologically irreducible bounded representations$(T, \calU)$of$G$on a Banach space$\calU$by giving a$G$-orbit in$\frn^*$($\frn$being the nilradical of$\frg$), a topologically irreducible representation of$L^1(\RR^n, \o)$, for a certain weight$\o$and a certain$n \in \NN$, and a topologically simple extension norm. If$G$is not symmetric, \ie, if the weight$\o$is exponential, we get a new type of representations which are fundamentally different from the induced representations. Soit$G$un groupe de Lie r\'esoluble exponentiel. Nous caract\'erisons toutes les repr\'esentations$(T, \calU)$continues born\'ees topologiquement irr\'eductibles de$G$dans un espace de Banach$\calU$\a l'aide d'une$G$-orbite dans$\frn^*$($\frn$\'etant le radical nilpotent de$\frg$), d'une repr\'esentation topologiquement irr\'eductible de$L^1(\RR^n, \o)$, pour un certain poids$\o$et un certain$n \in \NN$, d'une norme d'extension topologiquement simple. Si$G$n'est pas sym\'etrique, c. \a d. si le poids$\o$est exponentiel, nous obtenons un nouveau type de repr\'esentations qui sont fondamentalement diff\'erentes des repr\'esentations induites. Keywords:groupe de Lie rÃ©soluble exponentiel, reprÃ©sentation bornÃ©e topologiquement irrÃ©ductible, orbite, norme d'extension, sous-espace invariant, idÃ©al premier, idÃ©al primitifCategory:43A20 41. CJM 2001 (vol 53 pp. 565) Hare, Kathryn E.; Sato, Enji  Spaces of Lorentz Multipliers We study when the spaces of Lorentz multipliers from$L^{p,t} \rightarrow L^{p,s}$are distinct. Our main interest is the case when$s Keywords:multipliers, convolution operators, Lorentz spaces, Lorentz-improving multipliersCategories:43A22, 42A45, 46E30

42. CJM 2000 (vol 52 pp. 412)

Varopoulos, N. Th.
 Geometric and Potential Theoretic Results on Lie Groups The main new results in this paper are contained in the geometric Theorems 1 and~2 of Section~0.1 below and they are related to previous results of M.~Gromov and of myself (\cf\ \cite{1},~\cite{2}). These results are used to prove some general potential theoretic estimates on Lie groups (\cf\ Section~0.3) that are related to my previous work in the area (\cf\ \cite{3},~\cite{4}) and to some deep recent work of G.~Alexopoulos (\cf\ \cite{5},~\cite{21}). Categories:22E30, 43A80, 60J60, 60J65

43. 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

44. CJM 1999 (vol 51 pp. 96)

Rösler, Margit; Voit, Michael
 Partial Characters and Signed Quotient Hypergroups If $G$ is a closed subgroup of a commutative hypergroup $K$, then the coset space $K/G$ carries a quotient hypergroup structure. In this paper, we study related convolution structures on $K/G$ coming from deformations of the quotient hypergroup structure by certain functions on $K$ which we call partial characters with respect to $G$. They are usually not probability-preserving, but lead to so-called signed hypergroups on $K/G$. A first example is provided by the Laguerre convolution on $\left[ 0,\infty \right[$, which is interpreted as a signed quotient hypergroup convolution derived from the Heisenberg group. Moreover, signed hypergroups associated with the Gelfand pair $\bigl( U(n,1), U(n) \bigr)$ are discussed. Keywords:quotient hypergroups, signed hypergroups, Laguerre convolution, Jacobi functionsCategories:43A62, 33C25, 43A20, 43A90

45. CJM 1998 (vol 50 pp. 1090)

Lohoué, Noël; Mustapha, Sami
 Sur les transformÃ©es de Riesz sur les groupes de Lie moyennables et sur certains espaces homogÃ¨nes Let $\Delta$ be a left invariant sub-Laplacian on a Lie group $G$ and let $\nabla$ be the associated gradient. In this paper we investigate the boundness of the Riesz transform $\nabla\Delta^{-1/2}$ on Lie groups $G$ which are amenable and with exponential volume growth and on certain homogenous spaces. Categories:22E30, 35H05, 43A80, 43A85

46. CJM 1998 (vol 50 pp. 897)

Bloom, Walter R.; Xu, Zengfu
 Fourier multipliers for local hardy spaces on ChÃ©bli-TrimÃ¨che hypergroups In this paper we consider Fourier multipliers on local Hardy spaces $\qin$ $(0 Keywords:Fourier multipliers, Hardy spaces, hypergroupCategories:43A62, 43A15, 43A32 47. CJM 1997 (vol 49 pp. 1224) Ørsted, Bent; Zhang, Genkai  Tensor products of analytic continuations of holomorphic discrete series We give the irreducible decomposition of the tensor product of an analytic continuation of the holomorphic discrete series of$\SU(2, 2)$with its conjugate. Keywords:Holomorphic discrete series, tensor product, spherical function, Clebsch-Gordan coefficient, Plancherel theoremCategories:22E45, 43A85, 32M15, 33A65 48. CJM 1997 (vol 49 pp. 1117) Hu, Zhiguo  The von Neumann algebra$\VN(G)$of a locally compact group and quotients of its subspaces Let$\VN(G)$be the von Neumann algebra of a locally compact group$G$. We denote by$\mu$the initial ordinal with$\abs{\mu}$equal to the smallest cardinality of an open basis at the unit of$G$and$X= \{\alpha; \alpha < \mu \}$. We show that if$G$is nondiscrete then there exist an isometric$*$-isomorphism$\kappa$of$l^{\infty}(X)$into$\VN(G)$and a positive linear mapping$\pi$of$\VN(G)$onto$l^{\infty}(X)$such that$\pi\circ\kappa = \id_{l^{\infty}(X)}$and$\kappa$and$\pi$have certain additional properties. Let$\UCB (\hat{G})$be the$C^{*}$-algebra generated by operators in$\VN(G)$with compact support and$F(\hat{G})$the space of all$T \in \VN(G)$such that all topologically invariant means on$\VN(G)$attain the same value at$T$. The construction of the mapping$\pi$leads to the conclusion that the quotient space$\UCB (\hat{G})/F(\hat{G})\cap \UCB(\hat{G})$has$l^{\infty}(X)$as a continuous linear image if$G$is nondiscrete. When$G$is further assumed to be non-metrizable, it is shown that$\UCB(\hat{G})/F (\hat{G})\cap \UCB(\hat{G})$contains a linear isomorphic copy of$l^{\infty}(X)$. Similar results are also obtained for other quotient spaces. Categories:22D25, 43A22, 43A30, 22D15, 43A07, 47D35 49. CJM 1997 (vol 49 pp. 883) Okounkov, Andrei  Proof of a conjecture of Goulden and Jackson We prove an integration formula involving Jack polynomials conjectured by I.~P.~Goulden and D.~M.~Jackson in connection with enumeration of maps in surfaces. Categories:05E05, 43A85, 57M15 50. CJM 1997 (vol 49 pp. 736) Fendler, Gero  Dilations of one parameter Semigroups of positive Contractions on$L^{\lowercase {p}}$spaces It is proved in this note, that a strongly continuous semigroup of (sub)positive contractions acting on an$L^p$-space, for$1 Categories:47D03, 22D12, 43A22
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