26. CJM 2007 (vol 59 pp. 225)
27. CJM 2006 (vol 58 pp. 691)
 Bendikov, A.; SaloffCoste, L.

Hypoelliptic BiInvariant 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 

28. CJM 2006 (vol 58 pp. 768)
 Hu, Zhiguo; Neufang, Matthias

Decomposability of von Neumann Algebras and the Mazur Property of Higher Level
The decomposability
number of a von Neumann algebra $\m$ (denoted by $\dec(\m)$) is the
greatest cardinality of a family of pairwise orthogonal nonzero
projections in $\m$. In this paper, we explore the close
connection between $\dec(\m)$ and the cardinal level of the Mazur
property for the predual $\m_*$ of $\m$, the study of which was
initiated by the second author. Here, our main focus is on
those von Neumann algebras whose preduals constitute such
important Banach algebras on a locally compact group $G$ as the
group algebra $\lone$, the Fourier algebra $A(G)$, the measure
algebra $M(G)$, the algebra $\luc^*$, etc. We show that for
any of these von Neumann algebras, say $\m$, the cardinal number
$\dec(\m)$ and a certain cardinal level of the Mazur property of $\m_*$
are completely encoded in the underlying group structure. In fact,
they can be expressed precisely by two dual cardinal
invariants of $G$: the compact covering number $\kg$ of $G$ and
the least cardinality $\bg$ of an open basis at the identity of
$G$. We also present an application of the Mazur property of higher
level to the topological centre problem for the Banach algebra
$\ag^{**}$.
Keywords:Mazur property, predual of a von Neumann algebra, locally compact group and its cardinal invariants, group algebra, Fourier algebra, topological centre Categories:22D05, 43A20, 43A30, 03E55, 46L10 

29. 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 nonnegative.
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 

30. 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 operator Categories:43A85, 22D10, 39A70 

31. 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 

32. CJM 2005 (vol 57 pp. 17)
 Bédos, Erik; Conti, Roberto; Tuset, Lars

On Amenability and CoAmenability 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 coamenability for
such quantum groups. As a background for this study, we investigate
the associated tensor C$^{*}$categories.
Keywords:quantum group, amenability Categories:46L05, 46L65, 22D10, 22D25, 43A07, 43A65, 58B32 

33. 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 maps Category:43A32 

34. CJM 2004 (vol 56 pp. 344)
35. 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 

36. CJM 2003 (vol 55 pp. 1134)
 Casarino, Valentina

Norms of Complex Harmonic Projection Operators
In this paper we estimate the $(L^pL^2)$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 

37. 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 values Categories:43A90, 53C35, 15A18 

38. CJM 2002 (vol 54 pp. 1280)
 Skrzypczak, Leszek

Besov Spaces and Hausdorff Dimension For Some CarnotCarathÃ©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 CarnotCarath\'eodory metric on a
unimodular Lie group $G$. We define Besov spaces corresponding to the subLaplacian
$\Delta=\sum X_i^2$ 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, subelliptic operators, CarnotCarathÃ©odory metric, Hausdorff dimension Categories:46E35, 43A15, 28A78 

39. 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, projective Categories:13D03, 18G25, 43A95, 46L07, 22D99 

40. 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, frame Categories:42C40, 43A25, 42C15, 46N99 

41. 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 

42. CJM 2001 (vol 53 pp. 944)
 Ludwig, J.; MolitorBraun, 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, sousespace invariant, idÃ©al premier, idÃ©al primitif Category:43A20 

43. 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, Lorentzimproving multipliers Categories:43A22, 42A45, 46E30 

44. 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 

45. 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 

46. 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 probabilitypreserving, but lead to socalled 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 functions Categories:43A62, 33C25, 43A20, 43A90 

47. CJM 1998 (vol 50 pp. 1090)
48. CJM 1998 (vol 50 pp. 897)
49. 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, ClebschGordan coefficient, Plancherel theorem Categories:22E45, 43A85, 32M15, 33A65 

50. 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
nonmetrizable, 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 
