By Greg Kuperberg

ISBN-10: 0821853414

ISBN-13: 9780821853412

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

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Example text

46 3. EXAMPLES (b) Let V = {Vt } be a translation invariant quantum pseudometric on W ∗ (U , V ). 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.

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.

We verify the condition stated just before the proposition with the vectors vi and wi ranging over the standard basis {ex } of l2 (X). So let v1 , . . , vn , w1 , . . , wn be basis vectors and find a finite set S ⊆ X on which they are all supported. Fix 0 ≤ s < t. Then eventually we have dλ (x, y) ≤ t for all x, y ∈ S with d(x, y) ≤ s, so that if A ∈ [Vs ]1 then MχS AMχS ∈ [Vtλ ]1 , and |||A − MχS AMχS ||| = 0 for the seminorm ||| · |||. A similar argument shows that [Vsλ ]1 is eventually within the -neighborhood of [Vt ]1 for the seminorm ||| · |||, for all > 0.

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