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

By Greg Kuperberg

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Unless some degeneracy occurs such that L(φ) = L(φi ) = 0 for some i, we have ˜ L(φ1 ) = inf{C ≥ 0 : Vt1 ⊆ B(K)⊗V Ct for all t ≥ 0} = 1 2 L(φ2 ) = inf{C ≥ 0 : Vt ⊆ RVt1 R for all t ≥ 0} = 1 2 L(φ3 ) = inf{C ≥ 0 : Wt1 ⊆ VCt R for all t ≥ 0} ∗ 1 L(φ4 ) = inf{C ≥ 0 : Wt ⊆ U WCt U for all t ≥ 0} = 1 and L(φ) = L(φ3 ). Next we present three easy constructions. 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.

Then d((x, y), (x , y )) = inf{t : Mei(m(x−x )+n(y−y )) ∈ Vt } is a closed translation invariant pseudometric on T2 and V0 = {Vt0 } and V1 = {Vt1 } are translation invariant quantum pseudometrics on W ∗ (U , V ) where Vt0 Vt1 = VE0 (St ) = VE1 (St ) with St = {(x, y) ∈ T2 : d((0, 0), (x, y)) ≤ t}. 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 now turn to quotients, subobjects, and products. Quotients are simplest. If φ : M → N is a surjective unital weak* continuous ∗-homomorphism then ker(φ) is a weak* closed ideal of M, and hence ker(φ) = RM for some central projection R ∈ M. Thus M = RM ⊕ (I − R)M with (I − R)M ∼ = N . So metric quotients are modelled by von Neumann algebra direct summands. 35. Let V be a quantum pseudometric on a von Neumann algebra M ⊆ B(H). A metric quotient of M is a direct summand N = RM ⊆ B(K) of M, where R is a central projection in M and K = ran(R), together with the quantum pseudometric W = {Wt } on N defined by Wt = RVt R ⊆ B(K).

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