  We show that for algebraic groups over local fields of characteristic zero,
the following are equivalent: Every homomorphism has a closed image, every
unitary representation decomposes into a direct sum of finite-dimensional and
mixing representations, and that the matrix coefficients are dense within the
algebra of weakly almost periodic functions over the group. In our proof, we
employ methods from semi-group theory. We establish that algebraic groups are
\emph{compactification-centric}, meaning $sG = Gs$ for any element $s$ in the
weakly almost periodic compactification of the group $G$.
