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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

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End point compactification for metric spaces

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There is Noncommutative End Theory by Akemann and Eilers.

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