Let $I$ be a filtered poset, which you should think of as being huge. Let $A_i$ be an $I$-diagram of $C^{\star}$-algebras and let $A$ be the colimit of this diagram; if necessary, we can also assume that all structure maps $A_i \rightarrow A_j$ are inclusions. Given a nuclear $C^{\star}$-algebra $N$, is it then true that tensoring with $N$ commutes with the colimit, i.e. that $A \otimes N$ is the colimit of $A_i \otimes N$? In particular, I am interested in the case $N = C(X)$ for some compact space $X$.

I expect this to be true (and seem to have a proof for C(X) at least), but have not been able to find anything in the literature, so a reference would be welcome.

MUCHmore complicated $\endgroup$4more comments