1. CMB Online first
 Hare, Kathryn; Ramsey, L. Thomas

The relationship between $\epsilon$Kronecker sets and Sidon sets
A subset $E$ of a discrete abelian group is called $\epsilon
$Kronecker if
all $E$functions of modulus one can be approximated to within
$\epsilon $
by characters. $E$ is called a Sidon set if all bounded $E$functions
can be
interpolated by the Fourier transform of measures on the dual
group. As $%
\epsilon $Kronecker sets with $\epsilon \lt 2$ possess the same
arithmetic
properties as Sidon sets, it is natural to ask if they are Sidon.
We use the
Pisier net characterization of Sidonicity to prove this is true.
Keywords:Kronecker set, Sidon set Categories:43A46, 42A15, 42A55 

2. CMB 2015 (vol 58 pp. 632)
 Silberman, Lior

Quantum Unique Ergodicity on Locally Symmetric Spaces: the Degenerate Lift
Given a measure $\bar\mu_\infty$ on a locally symmetric space $Y=\Gamma\backslash
G/K$,
obtained as a weak{*} limit of probability measures associated
to
eigenfunctions of the ring of invariant differential operators,
we
construct a measure $\bar\mu_\infty$ on the homogeneous space $X=\Gamma\backslash
G$
which lifts $\bar\mu_\infty$ and which is invariant by a connected subgroup
$A_{1}\subset A$ of positive dimension, where $G=NAK$ is an Iwasawa
decomposition. If the functions are, in addition, eigenfunctions
of
the Hecke operators, then $\bar\mu_\infty$ is also the limit of measures
associated
to Hecke eigenfunctions on $X$. This generalizes results of the
author
with A. Venkatesh in the case where the spectral parameters
stay
away from the walls of the Weyl chamber.
Keywords:quantum unique ergodicity, microlocal lift, spherical dual Categories:22E50, 43A85 

3. CMB 2015 (vol 58 pp. 415)
 Willson, Benjamin

A Fixed Point Theorem and the Existence of a Haar Measure for Hypergroups Satisfying Conditions Related to Amenability
In this paper we present a fixed point property for amenable
hypergroups which is analogous to Rickert's fixed point theorem
for semigroups. It equates the existence of a left invariant
mean on the space of weakly right uniformly continuous functions
to the existence of a fixed point for any action of the hypergroup.
Using this fixed point property, a certain class of hypergroups
are shown to have a left Haar measure.
Keywords:invariant measure, Haar measure, hypergroup, amenability, function translations Categories:43A62, 43A05, 43A07 

4. CMB 2014 (vol 58 pp. 3)
5. CMB 2014 (vol 57 pp. 834)
 Koh, Doowon

Restriction Operators Acting on Radial Functions on Vector Spaces Over Finite Fields
We study $L^pL^r$ restriction estimates for
algebraic varieties $V$ in the case when restriction operators act on
radial functions in the finite field setting.
We show that if the varieties $V$ lie in odd dimensional vector
spaces over finite fields, then the conjectured restriction estimates
are possible for all radial test functions.
In addition, assuming that the varieties $V$ are defined in even
dimensional spaces and have few intersection points with the sphere
of zero radius, we also obtain the conjectured exponents for all
radial test functions.
Keywords:finite fields, radial functions, restriction operators Categories:42B05, 43A32, 43A15 

6. CMB 2013 (vol 57 pp. 449)
 Alaghmandan, Mahmood; Choi, Yemon; Samei, Ebrahim

ZLamenability Constants of Finite Groups with Two Character Degrees
We calculate the exact amenability constant of the centre of
$\ell^1(G)$ when $G$ is one of the following classes of finite group:
dihedral; extraspecial; or Frobenius with abelian complement and
kernel. This is done using a formula which applies to all finite
groups with two character degrees. In passing, we answer in the
negative a question raised in work of the third author with Azimifard
and Spronk (J. Funct. Anal. 2009).
Keywords:center of group algebras, characters, character degrees, amenability constant, Frobenius group, extraspecial groups Categories:43A20, 20C15 

7. CMB 2013 (vol 56 pp. 729)
 Currey, B.; Mayeli, A.

The Orthonormal Dilation Property for Abstract Parseval Wavelet Frames
In this work we introduce a class of discrete groups containing
subgroups of abstract translations and dilations, respectively. A
variety of wavelet systems can appear as $\pi(\Gamma)\psi$, where $\pi$ is
a unitary representation of a wavelet group and $\Gamma$ is the abstract
pseudolattice $\Gamma$. We prove a condition in order that a Parseval
frame $\pi(\Gamma)\psi$ can be dilated to an orthonormal basis of the
form $\tau(\Gamma)\Psi$ where $\tau$ is a superrepresentation of
$\pi$. For a subclass of groups that includes the case where the
translation subgroup is Heisenberg, we show that this condition
always holds, and we cite familiar examples as applications.
Keywords:frame, dilation, wavelet, BaumslagSolitar group, shearlet Categories:43A65, 42C40, 42C15 

8. CMB 2012 (vol 57 pp. 289)
 Ghasemi, Mehdi; Marshall, Murray; Wagner, Sven

Closure of the Cone of Sums of $2d$powers in Certain Weighted $\ell_1$seminorm Topologies
In a paper from 1976, Berg, Christensen and Ressel prove that the
closure of the cone of sums of squares $\sum
\mathbb{R}[\underline{X}]^2$ in the polynomial ring
$\mathbb{R}[\underline{X}] := \mathbb{R}[X_1,\dots,X_n]$ in the
topology induced by the $\ell_1$norm is equal to
$\operatorname{Pos}([1,1]^n)$, the cone consisting of all polynomials
which are nonnegative on the hypercube $[1,1]^n$. The result is
deduced as a corollary of a general result, established in the same
paper, which is valid for any commutative semigroup.
In later work, Berg and Maserick and Berg, Christensen and Ressel
establish an even more general result, for a commutative semigroup
with involution, for the closure of the cone of sums of squares of
symmetric elements in the weighted $\ell_1$seminorm topology
associated to an absolute value.
In the present paper we give a new proof of these results which is
based on Jacobi's representation theorem from 2001. At the same time,
we use Jacobi's representation theorem to extend these results from
sums of squares to sums of $2d$powers, proving, in particular, that
for any integer $d\ge 1$, the closure of the cone of sums of
$2d$powers $\sum \mathbb{R}[\underline{X}]^{2d}$ in
$\mathbb{R}[\underline{X}]$ in the topology induced by the
$\ell_1$norm is equal to $\operatorname{Pos}([1,1]^n)$.
Keywords:positive definite, moments, sums of squares, involutive semigroups Categories:43A35, 44A60, 13J25 

9. CMB 2012 (vol 57 pp. 37)
 Dashti, Mahshid; NasrIsfahani, Rasoul; Renani, Sima Soltani

Character Amenability of Lipschitz Algebras
Let ${\mathcal X}$ be a locally compact metric space and let
${\mathcal A}$ be any of the Lipschitz algebras
${\operatorname{Lip}_{\alpha}{\mathcal X}}$, ${\operatorname{lip}_{\alpha}{\mathcal X}}$ or
${\operatorname{lip}_{\alpha}^0{\mathcal X}}$. In this paper, we show, as a
consequence of rather more general results on Banach algebras,
that ${\mathcal A}$ is $C$character amenable if and only if
${\mathcal X}$ is uniformly discrete.
Keywords:character amenable, character contractible, Lipschitz algebras, spectrum Categories:43A07, 46H05, 46J10 

10. CMB 2011 (vol 56 pp. 13)
 Alon, Gil; Kozma, Gady

Ordering the Representations of $S_n$ Using the Interchange Process
Inspired by Aldous' conjecture for
the spectral gap of the interchange process and its recent
resolution by Caputo, Liggett, and Richthammer, we define
an associated order $\prec$ on the irreducible representations of $S_n$. Aldous'
conjecture is equivalent to certain representations being comparable
in this order, and hence determining the ``Aldous order'' completely is a
generalized question. We show a few additional entries for this order.
Keywords:Aldous' conjecture, interchange process, symmetric group, representations Categories:82C22, 60B15, 43A65, 20B30, 60J27, 60K35 

11. CMB 2011 (vol 56 pp. 218)
 Yang, Dilian

Functional Equations and Fourier Analysis
By exploring the relations among functional equations, harmonic analysis and representation theory,
we give a unified and very accessible approach to solve three important functional equations 
the d'Alembert equation, the Wilson equation, and the d'Alembert long equation 
on compact groups.
Keywords:functional equations, Fourier analysis, representation of compact groups Categories:39B52, 22C05, 43A30 

12. CMB 2011 (vol 54 pp. 654)
 Forrest, Brian E.; Runde, Volker

Norm One Idempotent $cb$Multipliers with Applications to the Fourier Algebra in the $cb$Multiplier Norm
For a locally compact group $G$, let $A(G)$ be its Fourier algebra, let $M_{cb}A(G)$ denote the completely
bounded multipliers of $A(G)$, and let $A_{\mathit{Mcb}}(G)$ stand for the closure of $A(G)$ in $M_{cb}A(G)$. We
characterize the norm one idempotents in $M_{cb}A(G)$: the indicator function of a set $E \subset G$ is a norm
one idempotent in $M_{cb}A(G)$ if and only if $E$ is a coset of an open subgroup of $G$. As applications, we
describe the closed ideals of $A_{\mathit{Mcb}}(G)$ with an approximate identity bounded by $1$, and we characterize
those $G$ for which $A_{\mathit{Mcb}}(G)$ is $1$amenable in the sense of B. E. Johnson. (We can even slightly
relax the norm bounds.)
Keywords:amenability, bounded approximate identity, $cb$multiplier norm, Fourier algebra, norm one idempotent Categories:43A22, 20E05, 43A30, 46J10, 46J40, 46L07, 47L25 

13. CMB 2011 (vol 54 pp. 663)
 Haas, Ruth; G. Helminck, Aloysius

Admissible Sequences for Twisted Involutions in Weyl Groups
Let $W$ be a Weyl group, $\Sigma$ a set of simple reflections in $W$
related to a basis $\Delta$ for the root system $\Phi$ associated with
$W$ and $\theta$ an involution such that $\theta(\Delta) = \Delta$. We
show that the set of $\theta$twisted involutions in $W$,
$\mathcal{I}_{\theta} = \{w\in W \mid \theta(w) = w^{1}\}$ is in one
to one correspondence with the set of regular involutions
$\mathcal{I}_{\operatorname{Id}}$. The elements of $\mathcal{I}_{\theta}$ are
characterized by sequences in $\Sigma$ which induce an ordering called
the RichardsonSpringer Poset. In particular, for $\Phi$ irreducible,
the ascending RichardsonSpringer Poset of $\mathcal{I}_{\theta}$,
for nontrivial $\theta$ is identical to the descending
RichardsonSpringer Poset of $\mathcal{I}_{\operatorname{Id}}$.
Categories:20G15, 20G20, 22E15, 22E46, 43A85 

14. CMB 2011 (vol 54 pp. 544)
15. CMB 2010 (vol 54 pp. 207)
 Chen, Jiecheng; Fan, Dashan

A Bilinear Fractional Integral on Compact Lie Groups
As an analog of a wellknown theorem on the bilinear
fractional integral on $\mathbb{R}^{n}$ by Kenig and Stein,
we establish the similar boundedness
property for a bilinear fractional integral on a compact Lie group. Our
result is also a generalization of our recent theorem
about the
bilinear fractional integral on torus.
Keywords:bilinear fractional integral, $L^p$ spaces, Heat kernel Categories:43A22, 43A32, 43B25 

16. CMB 2010 (vol 54 pp. 126)
17. CMB 2010 (vol 54 pp. 3)
 Bakonyi, M.; Timotin, D.

Extensions of Positive Definite Functions on Amenable Groups
Let $S$ be a subset of an amenable group $G$ such that $e\in S$ and
$S^{1}=S$. The main result of this paper states that if the Cayley
graph of $G$ with respect to $S$ has a certain combinatorial property,
then every positive definite operatorvalued function on $S$ can be
extended to a positive definite function on $G$. Several known
extension results are obtained as corollaries. New applications are
also presented.
Categories:43A35, 47A57, 20E05 

18. CMB 2010 (vol 53 pp. 491)
 Jizheng, Huang; Liu, Heping

The Weak Type (1,1) Estimates of Maximal Functions on the Laguerre Hypergroup
In this paper, we discuss various maximal functions on the Laguerre hypergroup $\mathbf{K}$ including the heat maximal function, the Poisson maximal function, and the HardyLittlewood maximal function which is consistent with the structure of hypergroup of $\mathbf{K}$. We shall establish the weak type $(1,1)$ estimates for these maximal functions. The $L^p$ estimates for $p>1$ follow from the interpolation. Some applications are included.
Keywords:Laguerre hypergroup, maximal function, heat kernel, Poisson kernel Categories:42B25, 43A62 

19. CMB 2010 (vol 53 pp. 447)
 Choi, Yemon

Injective Convolution Operators on l^{∞}(Γ) are Surjective
Let $\Gamma$ be a discrete group and let $f \in \ell^{1}(\Gamma)$. We observe that if the natural convolution operator $\rho_f: \ell^{\infty}(\Gamma)\to \ell^{\infty}(\Gamma)$ is injective, then $f$ is invertible in $\ell^{1}(\Gamma)$. Our proof simplifies and generalizes calculations in a preprint of Deninger and Schmidt by appealing to the direct finiteness of the algebra $\ell^{1}(\Gamma)$. We give simple examples to show that in general one cannot replace $\ell^{\infty}$ with $\ell^{p}$, $1\leq p< \infty$, nor with $L^{\infty}(G)$ for nondiscrete $G$. Finally, we consider the problem of extending the main result to the case of weighted convolution operators on $\Gamma$, and give some partial results.
Categories:43A20, 46L05, 43A22 

20. CMB 2008 (vol 51 pp. 60)
 Janzen, David

F{\o}lner Nets for Semidirect Products of Amenable Groups
For unimodular semidirect products of locally compact amenable
groups $N$ and $H$, we show that one can always construct a
F{\o}lner net of the form $(A_\alpha \times B_\beta)$ for $G$, where
$(A_\alpha)$ is a strong form of F{\o}lner net for $N$ and
$(B_\beta)$ is any F{\o}lner net for $H$. Applications to the
Heisenberg and Euclidean motion groups are provided.
Categories:22D05, 43A07, 22D15, 43A20 

21. CMB 2007 (vol 50 pp. 291)
22. CMB 2007 (vol 50 pp. 56)
 Gourdeau, F.; Pourabbas, A.; White, M. C.

Simplicial Cohomology of Some Semigroup Algebras
In this paper, we investigate the higher simplicial cohomology
groups of the convolution algebra $\ell^1(S)$ for various semigroups
$S$. The classes of semigroups considered are semilattices, Clifford
semigroups, regular Rees semigroups and the additive semigroups of
integers greater than $a$ for some integer $a$. Our results are of
two types: in some cases, we show that some cohomology groups are $0$,
while in some other cases, we show that some cohomology groups are
Banach spaces.
Keywords:simplicial cohomology, semigroup algebra Category:43A20 

23. CMB 2006 (vol 49 pp. 549)
 Führ, Hartmut

HausdorffYoung Inequalities for Group Extensions
This paper studies HausdorffYoung inequalities for certain group extensions,
by use of Mackey's theory. We consider the case in which the dual
action of the quotient group is free almost everywhere. This
result applies in particular to yield a HausdorffYoung inequality for
nonunimodular groups.
Categories:43A30, 43A15 

24. CMB 2004 (vol 47 pp. 389)
25. CMB 2004 (vol 47 pp. 475)
 Wade, W. R.

Uniqueness of Almost Everywhere Convergent Vilenkin Series
D. J. Grubb [3] has shown that uniqueness holds, under a
mild growth condition, for Vilenkin series which converge almost
everywhere to zero. We show that, under even less restrictive
growth conditions, one can replace the limit function 0 by an
arbitrary $f\in L^q$, when $q>1$.
Categories:43A75, 42C10 
