For a locally compact Hausdorff semigroup $S$, the $L^{\infty}$-representation algebra $\mathfrak{R}(S)$ was extensively studied by Dunkl and Ramirez. In this paper we give a characterization of the Banach algebra $\mathfrak{R}(S)$ of a foundation semigroup $S$ and as an application we determine some BSE semigroup algerbras.

In this note, we construct an infinite family of elliptic curves $E$ defined over $\mathbf{Q}$ whose Mordell-Weil group $E(\mathbf{Q})$ has rank exactly two under the parity conjecture.