Semi-group 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 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 $G$.
- Publication:
-
arXiv e-prints
- 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