# Download e-book for kindle: A von Neumann algebra approach to quantum metrics. Quantum by Greg Kuperberg

By Greg Kuperberg

ISBN-10: 0821853414

ISBN-13: 9780821853412

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

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**Additional resources for A von Neumann algebra approach to quantum metrics. Quantum relations**

**Sample text**

Let M ⊆ B(H) be a von Neumann algebra. If V is a quantum pseudometric on M then ρV (P, Q) = inf{D(A) : A ∈ B(H) and P (A ⊗ I)Q = 0} (with inf ∅ = ∞) is a quantum distance function on M; conversely, if ρ is a quantum distance function on M then Vρ = {Vtρ } [email protected]ρ Vtρ = {A ∈ B(H) : ρ(P, Q) > t ⇒ P (A ⊗ I)Q = 0} is a quantum pseudometric on M. The two constructions are inverse to each other. Proof. Let V be a quantum pseudometric on M. 8. Now let ρ be any quantum distance function. 7 since ρ(P, Q) > t for all A ∈ M .

1. A quantum metric for which Vt = {A ∈ B(H) : ρ(P, Q) > t ⇒ P AQ = 0}, with P and Q ranging over projections in M (cf. 10), and furthermore diam(V) > sup{ρ(P, Q) : P and Q are nonzero projections in M} (cf. 16). Let A be a dual operator system properly contained in B(H) such that for any nonzero v, w ∈ H there exists A ∈ A with Aw, v = 0. For instance, we could take A = {A ∈ B(H) : tr(AB) = 0} 37 38 3. EXAMPLES where B is any nonzero traceless self-adjoint trace class operator. (Suppose Aw, v = 0 for all A ∈ A, with v and w nonzero.

L, if and only if ρ(P, Q) = ∞ for some pair of linkable projections P and Q. This settles the case t = ∞; for the rest of the proof assume t is ﬁnite. Suppose t ∈ L ∪ R. Then Vt = V

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

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