I have been having some trouble with an exercise from Murphy's "$C^*$-Algebras and Operator Theory" recently. Chapter 6 Exercise 6 is as follows:
If $A$ is an AF-Algebra (i.e. a direct limit of finite-dimensional $C^*$-Algebras), then so is $M_n(A)$ for every $n \in \mathbb{N}$.
I was thinking one would have to show that the limit and the operation turning $A$ into the matrix algebra of itself commute, maybe by setting up a map between the two spaces and showing it is a $*$-isomorphism. I am not really sure how to set this up though, and I'm also not sure if it is even enough.
I was also wondering if this statement can be generalized to any direct limit. If $A$ is a direct limit of $C^*$-Algebras, is $M_n(A)$ one as well?
I guess in the general case one could try verifying the universal property. Is anything simplified in the AF case? I'm not really used to proofs involving category theory or universal properties, so I was wondering if there is a reason Murphy chose to frame the exercise this way.