TITLE: Do direct limits (filtered colimits) commute with pullbacks, in C*-algebras?
QUESTION [11 upvotes]: I asked this question already on math.stackexchange, 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.

REPLY [12 votes]: 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) $\mathcal{B}$.
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