Corollary 46. For every >0 there is δ>0 , independent of the algebras, their representations, and their Hilbert spaces, such that unital von Neumann algebras M,N⊆(H) with the same identity and d(M,N)<δ satisfy (M,N)<, d(M',N')<.

Choose δ from Theorem 1.1 ([Strong Kadison--Kastler stability]) with unitary displacement /2 . The resulting unitary satisfies uMu^*=N and u-I</2 . Conjugation by its diagonal amplification identifies the unit balls of M_n(M) and M_n(N) and moves each contraction by at most 2u-I , independently of n . The same estimate applies to the commutants because uM'u^*=N' .