1. CJM 2014 (vol 67 pp. 795)
 Di Nasso, Mauro; Goldbring, Isaac; Jin, Renling; Leth, Steven; Lupini, Martino; Mahlburg, Karl

On a Sumset Conjecture of ErdÅs
ErdÅs conjectured that for any set $A\subseteq \mathbb{N}$
with positive
lower asymptotic density, there are infinite sets $B,C\subseteq
\mathbb{N}$
such that $B+C\subseteq A$. We verify ErdÅs' conjecture in
the case that $A$ has Banach density exceeding $\frac{1}{2}$.
As a consequence, we prove that, for $A\subseteq \mathbb{N}$
with
positive Banach density (a much weaker assumption than positive
lower density), we can find infinite $B,C\subseteq \mathbb{N}$
such
that $B+C$ is contained in the union of $A$ and a translate of
$A$. Both of the aforementioned
results are generalized to arbitrary countable
amenable groups. We also provide a positive solution to ErdÅs'
conjecture for subsets of the natural numbers that are pseudorandom.
Keywords:sumsets of integers, asymptotic density, amenable groups, nonstandard analysis Categories:11B05, 11B13, 11P70, 28D15, 37A45 

2. CJM 2009 (vol 61 pp. 382)
 Miao, Tianxuan

Unit Elements in the Double Dual of a Subalgebra of the Fourier Algebra $A(G)$
Let $\mathcal{A}$ be a Banach algebra with a bounded right
approximate identity and let $\mathcal B$ be a closed ideal of
$\mathcal A$. We study the relationship between the right identities
of the double duals ${\mathcal B}^{**}$ and ${\mathcal A}^{**}$ under
the Arens product. We show that every right identity of ${\mathcal
B}^{**}$ can be extended to a right identity of ${\mathcal A}^{**}$ in
some sense. As a consequence, we answer a question of Lau and
\"Ulger, showing that for the Fourier algebra $A(G)$ of a locally
compact group $G$, an element $\phi \in A(G)^{**}$ is in $A(G)$ if and
only if $A(G) \phi \subseteq A(G)$ and $E \phi = \phi $ for all right
identities $E $ of $A(G)^{**}$. We also prove some results about the
topological centers of ${\mathcal B}^{**}$ and ${\mathcal A}^{**}$.
Keywords:Locally compact groups, amenable groups, Fourier algebra, identity, Arens product, topological center Category:43A07 

3. CJM 2004 (vol 56 pp. 344)