MF group

In mathematics, an MF group (standing for matricial field group) is a countable discrete group which can be approximated by finite-dimensional unitary groups using the operator norm. A group $\mathrm{\Gamma }$ is MF if it embeds into $$U\left(\frac{\prod _k{M}_{n_k}(\mathbb{ℂ})}{\underset{k}{⨁}{M}_{n_k}(\mathbb{ℂ})}\right)$$ for a sequence of positive integers $(n_k)$, where $M_n(\mathbb{ℂ})$ is the algebra of $n\times n$ complex matrices, and sequences whose operator norms tend to zero compose the denominator. Also, finite parts of the group allow for maps to unitary matrices which are approximately multiplicative and which keep group elements a fixed distance apart in operator norm. All groups that are locally embeddable into finite groups are MF. All free groups are also MF. A group being MF is the same as the group C*-algebra having quasidiagonality, for amenable groups. Every amenable group is MF.

No non-MF group is known to exist, and it is an open question whether every group is MF.

References

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