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I am trying to understand ideals of direct limits in the category of $C^{\ast}$-algebras.

Let $(A_n,f_n)$ be a direct sequence of $C^{\ast}$-algebras and let $I$ be a primitive (modular) ideal of direct limit $\varinjlim A_n $. Is it true that there exists primitive (modular) ideals $I_i$ of $A_i$ such that $I= \varinjlim I_i$?

Unfortunately I could not find any reference.Any reference or ideas?

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    $\begingroup$ No. Let $B$ be a simple unital AF C*-algebra that is not ultrasimplicial (that is, its ordered K$_0$ group is not of rank one); we can find examples so that in addition, $B$ is the completion of $A = \lim A_i$ ($A_i$ finite-dimensional C*-algebras) and the induced maps $K_0 (A_i) \to K_0(A_{i+1})$ are one to one (not all simple AF C*-algebras can be so realized, but lots can). Then $\{0\}$ is a primitive ideal, but cannot be a limit of primitive ideals in the $A_i$, since it can only be a limit of the zero ideals, all but finitely many of which are not primitive. $\endgroup$
    – David Handelman
    Commented May 31, 2020 at 14:31

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