Behaviour of direct limit with matrices

Question

I am trying to understand direct limits in the category of $C^*$-algebras by self reading. My last question was also related to direct limits. Here is another of my doubts:

Let $(A_n,f_n)$ be a direct sequence of $C^*$-algebras. Does the direct limit behave well with matrices i.e. $$\lim_{\rightarrow} M_2(A_n)=M_2(\lim_{\rightarrow} A_n)$$ Where for the system $M_2(A_n)$ the connecting maps are the natural maps obtained using $f_n$ componentwise.

I do feel like the result should be true, but I don’t really have an argument. Any ideas?

Answer 1

You’re asking whether the functor $M_2$ on Banach spaces preserves colimits of direct sequences.

(In case you’re not familiar with the categorical terminology I’m using here, don’t be put off — it’s not very deep, it’s just useful packaging-up of the kind of properties you’re discussing in the question, and hopefully this answer will give an idea of why this sort of categorical language is helpful.)

The forgetful functor $U$ from C*-algebras to Banach spaces preserves and reflects limits, so it’s enough to check the isomorphism $\varinjlim M_2(A_n) \cong M_2(\varinjlim A_n)$ on the underlying Banach spaces.

But at the level of Banach spaces, $U(M_2(A)) \cong U(A) \oplus U(A) \oplus U(A) \oplus U(A)$ — i.e. the coproduct of 4 copies of $U(A)$. So it’s enough to show that the functor $X \mapsto X \oplus X \oplus X \oplus X$ on Banach spaces preserves colimits of direct sequences.

But this is automatic, because colimits commute with colimits: concretely, check the universal properties a sequential colimit of coproducts, versus a coproduct of sequential colimits, and notice that they agree.