I [asked this question already on math.stackexchange](https://math.stackexchange.com/questions/869839), but maybe it is also useful to ask this here, since it was not answered there.


Suppose we have three directed sequences of $C^\*$-algebras, say $(A\_n,\varphi\_n)$,$(B\_n,\psi\_n)$ and $(C\_n,\theta\_n)$ and $\*$-homomorphisms $\alpha\_n:A\_n\rightarrow C\_n$ and $\beta\_n:B\_n\rightarrow C\_n$, then we can take the pullback $A\_n\times\_{C\_n}B\_n$ for all $n\in\mathbb{N}$ and can also take the direct limit, thus $\lim\_{n\rightarrow\infty}{A\_n\times\_{C\_n}B\_n}$. My question is: Does the following hold:
$$\lim\_{\rightarrow}{A\_n\times\_{C\_n}B\_n}=\lim\_{\rightarrow}{A\_n}\times\_{\lim\_{\rightarrow}{C\_n}}\lim\_{\rightarrow}{B\_n}$$
or in other words: do direct limits preserve pullbacks? From my point of view this does not hold in general but I can not find a good argument. Someone an idea? Thank you very much.



Do direct limits (filtered colimits) commute with pullbacks, in C*-algebras?
I may be showing my ignorance of category theory, but don't think this is true. Work on $l^2$. Let $\mathcal{A}\_n$ consist of the operators $A$ satisfying $\langle Ae\_i, e\_j\rangle = 0$ if $\max(i,j) > n$ (so $\mathcal{A}\_n$ is isomorphic to the $n\times n$ matrices). The direct limit of the $\mathcal{A}\_n$ is the compact operators on $l^2$. Let $P$ be a rank one projection that is not contained in any of the $\mathcal{A}\_n$, let $\mathcal{B} = \mathbb{C}\cdot P$ be the C\*-algebra it generates, and let $\mathcal{B}\_n = \mathcal{B}$ for all $n$. Also let $\mathcal{C}\_n = B(l^2)$ for all $n$. All the morphisms are inclusion maps.


Each pullback $\mathcal{A}\_n\times\_{\mathcal{C}\_n} \mathcal{B}\_n$ is zero, so the direct limit of the pullbacks is zero, but the pullback of the direct limits is (isomorphic to) $\mathcal{B}$.


