Semigroup compactifications of Algebraic Groups
Abstract
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 finitedimensional 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 semigroup theory. We establish that algebraic groups are \emph{compactificationcentric}, meaning $sG = Gs$ for any element $s$ in the weakly almost periodic compactification of the group $G$.
 Publication:

arXiv eprints
 Pub Date:
 April 2024
 DOI:
 10.48550/arXiv.2404.09878
 arXiv:
 arXiv:2404.09878
 Bibcode:
 2024arXiv240409878L
 Keywords:

 Mathematics  Group Theory;
 20G05;
 22E50;
 20M99;
 20G25;
 43A99;
 54H11;
 54H13;
 54D25