26. 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 

27. 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 

28. 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 

29. 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 

30. CMB 2007 (vol 50 pp. 291)
31. 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 

32. 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 

33. CMB 2004 (vol 47 pp. 445)
 Pirkovskii, A. Yu.

Biprojectivity and Biflatness for Convolution Algebras of Nuclear Operators
For a locally compact group $G$, the convolution product on
the space $\nN(L^p(G))$ of nuclear operators was defined by Neufang
\cite{Neuf_PhD}. We study homological properties of the convolution algebra
$\nN(L^p(G))$ and relate them to some properties of the group $G$,
such as compactness, finiteness, discreteness, and amenability.
Categories:46M10, 46H25, 43A20, 16E65 

34. 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 

35. CMB 2004 (vol 47 pp. 389)
36. CMB 2004 (vol 47 pp. 168)
 Baake, Michael; Sing, Bernd

Kolakoski$(3,1)$ Is a (Deformed) Model Set
Unlike the (classical) Kolakoski sequence on the alphabet $\{1,2\}$, its analogue
on $\{1,3\}$ can be related to a primitive substitution rule. Using this connection,
we prove that the corresponding biinfinite fixed point is a regular generic model set
and thus has a pure point diffraction spectrum. The Kolakoski$(3,1)$ sequence is
then obtained as a deformation, without losing the pure point diffraction property.
Categories:52C23, 37B10, 28A80, 43A25 

37. CMB 2004 (vol 47 pp. 215)
 Jaworski, Wojciech

Countable Amenable Identity Excluding Groups
A discrete group $G$ is called \emph{identity excluding\/}
if the only irreducible
unitary representation of $G$ which weakly contains the $1$dimensional identity
representation is the $1$dimensional identity representation itself. Given a
unitary representation $\pi$ of $G$ and a probability measure $\mu$ on $G$, let
$P_\mu$ denote the $\mu$average $\int\pi(g) \mu(dg)$. The goal of this article
is twofold: (1)~to study the asymptotic behaviour of the powers $P_\mu^n$, and
(2)~to provide a characterization of countable amenable identity excluding groups.
We prove that for every adapted probability measure $\mu$ on an identity excluding
group and every unitary representation $\pi$ there exists and orthogonal projection
$E_\mu$ onto a $\pi$invariant subspace such that $s$$\lim_{n\to\infty}\bigl(P_\mu^n
\pi(a)^nE_\mu\bigr)=0$ for every $a\in\supp\mu$. This also remains true for suitably
defined identity excluding locally compact groups. We show that the class of countable
amenable identity excluding groups coincides with the class of $\FC$hypercentral
groups; in the finitely generated case this is precisely the class of groups of
polynomial growth. We also establish that every adapted random walk on a countable
amenable identity excluding group is ergodic.
Categories:22D10, 22D40, 43A05, 47A35, 60B15, 60J50 

38. CMB 2003 (vol 46 pp. 268)
 Puls, Michael J.

Group Cohomology and $L^p$Cohomology of Finitely Generated Groups
Let $G$ be a finitely generated, infinite group, let $p>1$, and let
$L^p(G)$ denote the Banach space $\{ \sum_{x\in G} a_xx \mid \sum_{x\in
G} a_x ^p < \infty \}$. In this paper we will study the first
cohomology group of $G$ with coefficients in $L^p(G)$, and the first
reduced $L^p$cohomology space of $G$. Most of our results will be for a
class of groups that contains all finitely generated, infinite nilpotent
groups.
Keywords:group cohomology, $L^p$cohomology, central element of infinite order, harmonic function, continuous linear functional Categories:43A15, 20F65, 20F18 

39. CMB 2002 (vol 45 pp. 483)
 Baake, Michael

Diffraction of Weighted Lattice Subsets
A Dirac comb of point measures in Euclidean space with bounded
complex weights that is supported on a lattice $\varGamma$ inherits
certain general properties from the lattice structure. In
particular, its autocorrelation admits a factorization into a
continuous function and the uniform lattice Dirac comb, and its
diffraction measure is periodic, with the dual lattice
$\varGamma^*$ as lattice of periods. This statement remains true
in the setting of a locally compact Abelian group whose topology
has a countable base.
Keywords:diffraction, Dirac combs, lattice subsets, homometric sets Categories:52C07, 43A25, 52C23, 43A05 

40. CMB 2002 (vol 45 pp. 436)
41. CMB 2001 (vol 44 pp. 231)
 Rosenblatt, Joseph M.; Willis, George A.

Weak Convergence Is Not Strong Convergence For Amenable Groups
Let $G$ be an infinite discrete amenable group or a nondiscrete
amenable group. It is shown how to construct a net $(f_\alpha)$ of
positive, normalized functions in $L_1(G)$ such that the net converges weak*
to invariance but does not converge strongly to invariance. The solution of
certain linear equations determined by colorings of the Cayley graphs of the
group are central to this construction.
Category:43A07 

42. CMB 2000 (vol 43 pp. 330)
 Hare, Kathryn E.

Maximal Operators and Cantor Sets
We consider maximal operators in the plane, defined by Cantor sets of
directions, and show such operators are not bounded on $L^2$ if the
Cantor set has positive Hausdorff dimension.
Keywords:maximal functions, Cantor set, lacunary set Categories:42B25, 43A46 

43. CMB 1999 (vol 42 pp. 169)
 Ding, Hongming

Heat Kernels of Lorentz Cones
We obtain an explicit formula for heat kernels of Lorentz cones, a
family of classical symmetric cones. By this formula, the heat
kernel of a Lorentz cone is expressed by a function of time $t$ and
two eigenvalues of an element in the cone. We obtain also upper and
lower bounds for the heat kernels of Lorentz cones.
Keywords:Lorentz cone, symmetric cone, Jordan algebra, heat kernel, heat equation, LaplaceBeltrami operator, eigenvalues Categories:35K05, 43A85, 35K15, 80A20 

44. CMB 1998 (vol 41 pp. 392)
45. CMB 1997 (vol 40 pp. 316)
46. CMB 1997 (vol 40 pp. 296)
47. CMB 1997 (vol 40 pp. 183)
 Kepert, Andrew G.

The range of group algebra homomorphisms
A characterisation of the range of a homomorphism between two
commutative group algebras is presented which implies, among other
things, that this range is closed. The work relies mainly on the
characterisation of such homomorphisms achieved by P.~J.~Cohen.
Categories:43A22, 22B10, 46J99 

48. CMB 1997 (vol 40 pp. 133)
 Blackmore, T. D.

Derivations from totally ordered semigroup algebras into their duals
For a wellbehaved measure $\mu$, on a locally compact
totally ordered set $X$, with continuous part $\mu_c$, we make
$L^p(X,\mu_c)$
into a commutative Banach bimodule over the totally ordered
semigroup algebra
$L^p(X,\mu)$, in such a way that the natural surjection from the algebra
to the module is a bounded derivation. This gives rise to bounded
derivations from $L^p(X,\mu)$
into its dual module and in particular shows that if $\mu_c$ is not
identically zero then $L^p(X,\mu)$ is not weakly
amenable. We show that all bounded derivations from $L^1(X,\mu)$
into its dual module arise in this way and also describe all bounded
derivations from
$L^p(X,\mu)$ into its dual for $1
Categories:43A20, 46M20 
