Direct limit of totally ordered system of isomorphic $C^*$-algebras

Question

Suppose we have a directed system of $C^*$-algebras $A_i$

$$\{A_i,\phi_i\}_{i\in\mathbb{N}},$$

such that each $*$-homomorphism $\phi_i:A_i\rightarrow A_{i+1}$ is an inclusion. Furthermore, suppose that all of the $A_i$ are (abstractly) isomorphic.

Question: Is it true that the direct limit of this system is isomorphic to each $A_i$?

Answer 1

No. Take for instance $A_j=B(\ell^2(\mathbb N))$ for all $j$, and use embeddings $$ T\longmapsto \begin{bmatrix} T&0\\0&T\end{bmatrix}. $$ The direct limit has no minimal projections, so it cannot be isomorphic to $B(H)$.

If you want a separable example, you can use $K(\ell^2(\mathbb N))$ instead.