1. CMB Online first
 Pachl, Jan; Steprāns, Juris

Continuity of convolution and SIN groups
Let the measure algebra of a topological group $G$ be equipped
with
the topology of uniform convergence on bounded right uniformly
equicontinuous sets of functions.
Convolution is separately continuous on the measure algebra,
and it is jointly continuous if and only if $G$ has the SIN property.
On the larger space $\mathsf{LUC}(G)^\ast$ which includes the measure
algebra,
convolution is also jointly continuous if and only if the group
has the SIN property,
but not separately continuous for many nonSIN groups.
Keywords:topological group, SIN property, measure algebra, convolution Categories:43A10, 22A10 

2. CMB 2007 (vol 50 pp. 632)
 Zelenyuk, Yevhen; Zelenyuk, Yuliya

Transformations and Colorings of Groups
Let $G$ be a compact topological group and let $f\colon G\to G$ be a
continuous transformation of $G$. Define $f^*\colon G\to G$ by
$f^*(x)=f(x^{1})x$ and let $\mu=\mu_G$ be Haar measure on $G$. Assume
that $H=\Imag f^*$ is a subgroup of $G$ and for every
measurable $C\subseteq H$,
$\mu_G((f^*)^{1}(C))=\mu_H(C)$. Then for every measurable
$C\subseteq G$, there exist $S\subseteq C$ and $g\in G$ such that
$f(Sg^{1})\subseteq Cg^{1}$ and $\mu(S)\ge(\mu(C))^2$.
Keywords:compact topological group, continuous transformation, endomorphism, Ramsey theoryinversion, Categories:05D10, 20D60, 22A10 
