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By Greg Kuperberg

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32. (a) Let V be a quantum pseudometric on a von Neumann algebra M ⊆ B(H) and let C ≥ 0. Then the truncation of V to C is the quantum ˜ = (V˜t ) defined by pseudometric V V˜t = Vt B(H) if t < C if t ≥ C. (b) Let V and W be quantum pseudometrics on von Neumann algebras M ⊆ B(H) and N ⊆ B(K). Their direct sum is the von Neumann algebra M ⊕ N ⊆ B(H ⊕ K) equipped with the quantum pseudometric V ⊕ W = {Vt ⊕ Wt }. (c) Let {Vλ } with Vλ = {Vtλ } be a family of quantum pseudometrics on a von Neumann algebra M ⊆ B(H).

Conversely, Y = Y Pt for any Y ∈ It , so that ΦPt (B) = 0 implies ΦY (B) = ΦY ΦPt (B) = 0 for all Y ∈ It implies B ∈ Vt . Thus Vt = {B : ΦPt (B) = 0}. Finally, the Pt for t ≥ 0 constitute a decreasing right continous one-parameter family of projections in M ⊗ Mop and hence are the spectral projections P(t,∞) (X) for some positive operator X ∈ M ⊗ Mop . Now we proceed to our main result which gives a general intrinsic characterization of quantum pseudometrics. 7. 2). 17. 45. Let M ⊆ B(H) be a von Neumann algebra.

QUANTUM METRICS ˜ ∈ N ⊗B(l2 ). We set L(φ) = ∞ if φ is not co-Lipschitz. for all projections P˜ , Q Thus ρ(P, Q) L(φ) = sup P,Q ρ((φ ⊗ id)(P ), (φ ⊗ id)(Q)) with P and Q ranging over projections in M⊗B(l2 ) and using the convention 0 ∞ 0 = ∞ = 0. 28. Let V1 , V2 , and V3 be quantum pseudometrics on von Neumann algebras M1 , M2 , and M3 and let φ : M1 → M2 and ψ : M2 → M3 be co-Lipschitz morphisms. Then ψ ◦ φ : M1 → M3 is a co-Lipschitz morphism and L(ψ ◦ φ) ≤ L(ψ)L(φ). 27 is motivated by the atomic abelian case, where the unital weak* continuous ∗-homomorphisms from l∞ (X) to l∞ (Y ) are precisely the maps given by composition with functions from Y to X.

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