I think it should be important. Namely, it is supposed to be one of the major techniques to look at shape theory in noncommutative generalization (see e.g. papers by Dadarlat). In the commutative case, in shape theory, one approximates with inverse systems (not just sequences) of ANRs. I forgot which class of $C^\ast$-algebras they exactly corresponds in commutative case (there is a paper which does this). But I am rusty in this, I recall listening a comprehensive talk in Spring 2002 at Purdue by some collaborator of Dadarlat, this talk spurred my interest in UHF algebras and then I looked again few times few years ago. But now my memory is rusty. In any case, I think it deserves some care eventually. Hopefully I will have once a chance to work more at the entry and literature.

]]>True, the literature speaks of sequential colimits. Does it make a difference?

]]>Colimit may be indexed over any directed set here, not just a sequence ?

]]>stub for *uniformly hyperfinite algebra*