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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 $$d(M,N)=d(UM,UN)=d(MU,NU).$$

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    $\begingroup$ To construct the hyperfinite II$_1$ factor in this way you also need to use the uniform norm. You need to restrict to uniformly bounded subsets when you take the completion so that multiplication is continuous in the two norm. $\endgroup$
    – Jesse Peterson
    Commented Apr 30, 2013 at 16:08
  • $\begingroup$ What exactly do you mean by uniform norm? Do you mean you need to use a different norm on $M_n$ than the two norm/maximum singular value? $\endgroup$
    – J. E. Pascoe
    Commented May 1, 2013 at 4:00
  • $\begingroup$ For $A \in M_n$, the two norm is $\| A \|_2 = (\frac{1}{n} {\rm Tr}(A^*A) )^{1/2}$. This is different than the uniform norm/maximum singular value $\| A \|$. Although, one has the inequality $\| A \|_2 \leq \| A \|$. One way to construct the hyperfinite II$_1$ factor is to complete each uniform ball in the two norm and then take the union. But if you don't restrict to the uniform balls then you will not get an algebra since multiplication is not jointly continuous in the two norm $\endgroup$
    – Jesse Peterson
    Commented May 1, 2013 at 16:59

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It seems that you might like the (normalized) rank metric, producing the first algebraic examples of continuous geometries and their coordinatizing rings. See the original papers of von Neumann (1936 and 1937 in the PNAS) or Goodearl's book, von Neumann regular rings.

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