Science Mathematics

Download e-book for iPad: 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).

Show description

Read Online or Download A von Neumann algebra approach to quantum metrics. Quantum relations PDF

Best science & mathematics books

Download PDF by Jean-Pierre Serre, Catriona Maclean Pierre Colmez: Grothendieck-Serre Correspondence

This outstanding quantity encompasses a huge a part of the mathematical correspondence among A. Grothendieck and J-P. Serre. It kinds a brilliant advent to the advance of algebraic geometry in the course of the years 1955-1965. in this interval, algebraic geometry went via a awesome transformation, and Grothendieck and Serre have been between valuable figures during this procedure.

Wavelets: Algorithms & Applications by Yves Meyer, Robert D. Ryan PDF

During this textual content, the writer provides mathematical heritage and significant wavelet functions, starting from the electronic cell to galactic constitution and production of the universe. It discusses intimately the ancient origins, the algorithms and the purposes of wavelets.

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

Sample text

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 .

Download PDF sample

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

by Anthony

Rated 4.21 of 5 – based on 50 votes