Lemma 22 ([Norm control of tracial conjugations]). Let M,N⊆(K) be₁ factors in common standard tracial position with vector Ω . Suppose that R⊆M∩N is an irreducible hyperfinite subfactor of M , presented by increasing matrix algebras, and …

Lemma 21 ([Testing a bimodular operator on bounded algebra elements]). Let M be a₁ factor and let R=(⋃_{jQ_j})''⊆M be as in Lemma 18 ([Increasing matrix algebras with scalar relative commutant]). If a bounded operator V on L²(M,τ) …

Lemma 20 ([The common right action]). In the setting of Proposition 19 ([Common-position theorem]), the right actions of R agree elementwise: J_M rJ_M=J_N rJ_N (r∈R). In particular T=J_{NJ_M}-I commutes with the left and right actions of R .

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Set T=J_NJ_M-I . The common right-action identity implies that T commutes with both actions of R . For x∈M_1 choose y∈N_1 with x-y<b . Then TxΩ =J_Nx^*Ω-xΩ ≤(x^*-y^*)Ω+(x-y)Ω<2b. Thus the supremum in Lemma 21 ([Testing a bimodular operator on bounded algebra elements]) is at most 2b . That lemma gives T≤768b , and T=J_M-J_N .

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Lemma 22 ([Norm control of tracial conjugations]). Let M,N⊆(K) be1 factors in common standard tracial position with vector Ω . Suppose that R⊆M∩N is an irreducible hyperfinite subfactor of M , presented by increasing matrix algebras, and …

Lemma 21 ([Testing a bimodular operator on bounded algebra elements]). Let M be a1 factor and let R=(⋃jQj)''⊆M be as in Lemma 18 ([Increasing matrix algebras with scalar relative commutant]). If a bounded operator V on L2(M,τ) …

Lemma 20 ([The common right action]). In the setting of Proposition 19 ([Common-position theorem]), the right actions of R agree elementwise: JM rJM=JN rJN (r∈R). In particular T=JNJM-I commutes with the left and right actions of R .

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Lemma 22 ([Norm control of tracial conjugations]). Let M,N⊆(K) be₁ factors in common standard tracial position with vector Ω . Suppose that R⊆M∩N is an irreducible hyperfinite subfactor of M , presented by increasing matrix algebras, and …

Lemma 21 ([Testing a bimodular operator on bounded algebra elements]). Let M be a₁ factor and let R=(⋃_{jQ_j})''⊆M be as in Lemma 18 ([Increasing matrix algebras with scalar relative commutant]). If a bounded operator V on L²(M,τ) …

Lemma 20 ([The common right action]). In the setting of Proposition 19 ([Common-position theorem]), the right actions of R agree elementwise: J_M rJ_M=J_N rJ_N (r∈R). In particular T=J_{NJ_M}-I commutes with the left and right actions of R .