$\mathrm{MF}$-Property for Countable Discrete Groups

We say that a group has an $\mathrm{MF}$-property if it can be embedded in the group of unitary elements of the $C^*$-algebra $\prod M_n/\bigoplus M_n$. In the present paper we prove the $\mathrm{MF}$-property for the Baumslag group ${\langle a,b \mid a^{a^b}=a^2\rangle}$ and also some general assertions concerning this property.