Let $M_n$ denote the $n$ by $n$ matrices. Consider the homomorphisms $$\phi_{n,kn}: M_n \rightarrow M_{kn}$$ which takes a matrix $A \in M_n$ to $A \otimes I_k \in M_{kn}.$ This gives a sensible way to interpret the direct limit of $M_n.$ In the case where the base field is $\mathbb{C},$ the $\phi_{n,kn}$ are isometries in the two norm, and completing under the norm on this direct limit gives a special algebra, the hyperfinite $II_1$ factor.

What kind of (nondiscrete) metrics can we use to force $\phi_{n,kn}$ to be an isometry over general fields (maybe with some structural assumptions such as local) and what do the completions look like?

Specifically, I would prefer a metric such that for any $M$ there are isometries $U$ and $V$, and a diagonal matrix $D$ such that $$M = VDU.$$ Isometry is this case meaning $$d(M,N)=d(UM,UN)=d(MU,NU).$$