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

Proposition 24. There are absolute constants b0>0 and Cst >0 such that the following holds. Let M,N⊆(K) be type1 factors on a separable Hilbert space in common standard tracial position with vector Ω . Suppose they contain a common …

Lemma 23 ([The orientation of a Jordan map into a factor]). A surjective linear Jordan homomorphism between unital complex algebras, whose target is a factor von Neumann algebra, is either a homomorphism or an antihomomorphism.

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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.

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