Fixed point compactifications

Abstract

A fixed point compactification of a locally compact noncompact group G is a faithful semigroup compactification S such that $ap=pa=p$ for all $p\in S\setminus G$ and $a\in G$. Since the right translations are continuous, the remainder of a fixed point compactification is a right zero semigroup. Among all fixed point compactifications of G there is a largest one, denoted $\theta G$. We show that if G is $\sigma$-compact, then $\theta G\setminus G$ contains a copy of $\beta \omega \setminus \omega$. In contrast, if G is not $\sigma$-compact, then $\theta G$ is the one-point compactification.