By Greg Kuperberg

ISBN-10: 0821853414

ISBN-13: 9780821853412

Quantity 215, quantity 1010 (first of five numbers).

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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.

We have V0 ≤ V ≤ V1 . 16 (a) converges to the W*-ﬁltration Vr as → r. 14). The right notion of convergence seems to be the following. Denote the closed unit ball of any Banach space V by [V]1 . 17. Let {Vλ } be a net of W*-ﬁltrations of B(H). We say that {Vλ } locally converges to a W*-ﬁltration V of B(H) if for every 0 ≤ s < t and every weak* open neighborhood U of 0 ∈ B(H) we eventually have [Vsλ ]1 ⊆ [Vt ]1 + U and [Vs ]1 ⊆ [Vtλ ]1 + U. Equivalently, for any > 0 and any vectors v1 , . . , vn , w1 , .

26. Let V be a quantum pseudometric on a von Neumann algebra M ⊆ B(H). Then V is a quantum metric if and only if the closed projections in M⊗B(l2 ) generate M⊗B(l2 ) as a von Neumann algebra. Proof. Let N ⊆ M⊗B(l2 ) be the von Neumann algebra generated by the closed projections. , V is a quantum metric. Observe ﬁrst that every projection in I ⊗ B(l2 ) is closed. Thus N ⊆ (I ⊗ 2 B(l )) = B(H) ⊗ I. Now if A ∈ V0 then the range of any closed projection is clearly invariant for A ⊗ I. Since A∗ also belongs to V0 it follows that A ⊗ I commutes with every closed projection, and therefore V0 ⊗ I ⊆ N .

### A von Neumann algebra approach to quantum metrics. Quantum relations by Greg Kuperberg

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