existence of inductive limit of the inductive sequence of $C^*$-algebras

Question

Suppose $A_n$ is a sequnece of $C^*$-algebras and $h_n:A_n \to A_{n+1}$ is a sequence of homomorphisms,as it mentioned in Murphy's book,we can construct an inductive limit of $(A_n,h_n)$.

But in Rodam's book,there is a remark:inductive limits ,when they exist,are essentially unique .

Does there exist an inductive sequence which has no inductive limit?

Answer 1

Every inductive sequence of $C^*$-algebras has a limit, by Proposition 6.2.4 in Rørdam's book (the page after the one you quoted).