Lemma 25 ([Amplification by common isometries]). Let M,N⊆(H) contain the same unital copy of (ℓ²) . Then (M,N)=d(M,N) .

For each n≥1 choose isometries s_1,…,s_n in that common copy such that s_i^*s_j=δ_ ij 1 and ∑_i s_is_i^*=1 . The operator V:H^n→H given by V(ξ_i)=∑_i s_iξ_i is unitary. For either P=M or P=N , V M_n(P)V^*=P, V[x_ ij ]V^*=∑_ i,j s_ix_ ij s_j^*. Indeed the inverse coefficient formula is x_ ij =s_i^*xs_j . The same unitary therefore identifies the two amplified unit balls with the two original unit balls. Taking the supremum over n proves the assertion.