By Greg Kuperberg

ISBN-10: 0821853414

ISBN-13: 9780821853412

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

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**Example text**

For the converse, let the closure of q be q¯ = X − {p : ρ(p, q) > 0}. If ρ is not a measurable metric then the closed projections in L∞ (X, μ) do not 24 2. 5 of [35]). There must therefore exist an operator A ∈ B(L2 (X, μ)) that commutes with Mq for every closed projection q in L∞ (X, μ) but does not belong to M. Now if A ∈ V0ρ then there exist projections p, q ∈ L∞ (X, μ) with ρ(p, q) > 0 and Mp AMq = 0, but then A cannot commute with Mq¯, a contradiction. So we conclude that A ∈ V0ρ , and this shows that V0ρ = M.

Every metric quotient of M is of this form. 2), the von Neumann algebra ˜ = {Mf : f ∈ l∞ (X), x ∼ y ⇒ f (x) = f (y)} ⊆ M M equipped with the quantum pseudometric Vd˜ is a metric subobject of M. Every metric subobject of M is of this form. (c) The metric product of M and N is the von Neumann algebra M⊗N ∼ = l∞ (X × Y ) equipped with the quantum pseudometric VdX×Y associated to the pseudometric dX×Y ((x1 , y1 ), (x2 , y2 )) = max{d(x1 , x2 ), d (y1 , y2 )}. The proof is straightforward. 3). 6. Intrinsic characterization We will show that quantum pseudometrics can be characterized intrinsically in terms of quantum distance functions.

Then ρ(P˜ , Q) ˜ ≥ s but (P˜ , Q) ˜ ∈ R so ρ(φ(P˜ ), φ(Q)) ˜ ≤ t, P˜ Vs Q so we have ˜ ρ(P˜ , Q) s ≥ . ˜ ˜ t ρ(φ(P ), φ(Q)) Taking s → ρ(P, Q) and t → ρ((φ ⊗ id)(P ), (φ ⊗ id)(Q)) shows that ρ(P, Q) ˜ ≤ L(φ) ρ((φ ⊗ id)(P ), (φ ⊗ id)(Q)) ˜ and taking the supremum over P and Q ﬁnally yields L(φ) ≤ L(φ). 28 2. 6 but it has an equivalent version in terms of W*-ﬁltrations. 5) which states that every such map can be expressed as an inﬂation followed by a restriction followed by an isomorphism. Since this expression is not unique, if we deﬁned L(φ) in the concrete way indicated below then the deﬁnition would appear to be ambiguous.

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

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