Semi-group compactifications of Algebraic Groups

Abstract

We show that for connected affine algebraic groups over local fields of characteristic zero, the following are equivalent: every continuous homomorphism into a Hausdorff topological group has closed image, every unitary representation decomposes into a direct sum of finite-dimensional representations and representations that are mixing modulo their kernels, and the matrix coefficients are uniformly dense in the algebra of weakly almost periodic functions on the group. In our proof, we employ methods from semigroup theory. We establish that these groups are \emph{compactification-centric}, meaning $sG=Gs$ for every element $s$ in the weakly almost periodic compactification $w(G)$ of the group $G$.