By Greg Kuperberg

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We just give one illustration of the sufficiency of projections in M when V is reflexive; cf. 1. 19. 30 below. 24. Let V be a reflexive quantum pseudometric on a von Neumann algebra M ⊆ B(H). Then Vt = {A ∈ B(H) : ρ(P, Q) > t ⇒ P AQ = 0} for all t ∈ [0, ∞), with P and Q ranging over projections in M. Proof. Fix t and let V˜t = {A ∈ B(H) : ρ(P, Q) > t ⇒ P AQ = 0}. 6). Conversely, let A ∈ V˜t . For any s > t we have P Vs Q = 0 ⇒ ρ(P, Q) ≥ s ⇒ P AQ = 0 for any projections P, Q ∈ M. By reflexivity we conclude that A belongs to Vs for all s > t, and hence that A belongs to Vt .

Then wk∗ , the weak* closure of the algebraic product taken [t] times, set Vt = V · . . · V where [t] is the greatest integer ≤ t and with the convention that the empty product is M . We call V = {Vt } the quantum graph metric associated to V. We are most interested in the case where M = B(H), V0 = CI, and V is any dual operator system in B(H). 4. Let M ⊆ B(H) be a von Neumann algebra and let V be a dual operator system that is a bimodule over M . Then the quantum graph metric is the smallest quantum metric V on M such that V1 = V.

19. 30 below. 24. Let V be a reflexive quantum pseudometric on a von Neumann algebra M ⊆ B(H). Then Vt = {A ∈ B(H) : ρ(P, Q) > t ⇒ P AQ = 0} for all t ∈ [0, ∞), with P and Q ranging over projections in M. Proof. Fix t and let V˜t = {A ∈ B(H) : ρ(P, Q) > t ⇒ P AQ = 0}. 6). Conversely, let A ∈ V˜t . For any s > t we have P Vs Q = 0 ⇒ ρ(P, Q) ≥ s ⇒ P AQ = 0 for any projections P, Q ∈ M. By reflexivity we conclude that A belongs to Vs for all s > t, and hence that A belongs to Vt . Thus Vt = V˜t . Next we observe that reflexivity can always be achieved by stabilization.

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