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It is known that, using an idea of T. T. West, the unit ball of $L^{\infty}[0,1]$ can be identified as a compactification of $\mathbb{Z}$. In this talk, we will generalize this result to any locally compact Abelian group $G$. Depending on the algebraic properties of $G$, we will construct a semigroup compactification as a certain compact subsemigroup of $L^{\infty}[0,1]$, which is a quotient of both the Eberlein compactification, $G^e$, and the weakly almost periodic compactification, $G^w$, of $G$. The concrete structure of these compact quotients allows us to gain insight into known results by G. Brown, W. Moran, J. Pym and B. Bordbar, where for groups $G= \mathbb{Z}$ and $G=\mathbb{Z}_q^{\infty}$, it is proved that $G^w$ contains uncountably many idempotents and the set of idempotents is not closed.