A lemma on approximation of finite-dimensional $*$-algebras

The analogue of Glimm's lemma is proved: if the elements generating a finite-dimensional subalgebra $A$ of a finite type factor can be sufficiently well approximated by some elements of a subfactor $B$ they can also be approximated by elements of $B$ which generate an algebra isomorphic to $A$. The trace norm is used instead of the uniform one.