Fixed point compactifications

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.