I had seen somewhere in the literature that the Freudenthal compactification of a locally compact, connected, locally connected, $\sigma$-compact, Hausdorff topological space $X$, is the maximal ideal space of the $C^*$-algebra of the bounded continuous functions $f:X\rightarrow \mathbb{R}$ with the property that there is a compact subset $K\subseteq X$ such that that $f(X\setminus K)$ is a finite set. However, that paper seems to vanish into the thin air, and I don't remember the author's name of that paper. I will be grateful if someone could help me find that paper.
Update: While continuous functions with the above property can be continuously extended to the Freudenthal compactification, the converse is not true. For example, the Freudenthal compactification of $X=(0,1)$ is the closed interval $[0,1]$, take the identity function $\mathrm{id}:[0,1]\rightarrow [0,1]$, then there is no compact subset $K\subseteq X$ such that that $(X\setminus K)$ is a finite set.
