We know that every compactification of locally compact Hausdorff spaces correspond to a unitization of $C^{\*}$ algebras. For example the one point compactification corresponds to the minimal unitization and the Stone Chech compactification correspond to the maximal unitization "Multiplier algebra".


In this question I would like to know if something is already known for noncommutative analogy of "End points compactification" as described bellows?:(Your answers and comments are very appreciated) 


<http://en.wikipedia.org/wiki/End_%28topology%29>


and


[End point compactification for metric spaces](https://mathoverflow.net/questions/42961/end-point-compactification-for-metric-spaces)



Unitization via "End points compactification"
There is [**Noncommutative End Theory**](http://www.msp.org/pjm/1998/185-1/pjm-v185-n1-p03-p.pdf) by Akemann and Eilers.


