I am trying to understand direct limit in category of $C^\*$ algebras. 



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> Is it well known that direct limit behaves well with double dual and quotient of $C^\*$ algebras?
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Any references or ideas?


P.S: This question was first asked on mathatack but there I did not get any answer!



Behaviour of Direct limit with quotient and double dual
It definitely does not work for double duals. Indeed, if you view compact operators as an inductive limit of matrix algebras, then the sequence of double duals is exactly the same, but the double dual of compacts is $B(H)$.


