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By Greg Kuperberg

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A similar argument shows that [Vsλ ]1 is eventually within the -neighborhood of [Vt ]1 for the seminorm ||| · |||, for all > 0. We conclude that Vλ locally converges to V. 6. HOLDER METRICS 47 Now we show that translation invariant quantum pseudometrics on quantum torus von Neumann algebras converge as the parameter varies. 19. 16 (a)). Let r ∈ R. Then V locally converges to Vr as → r. Proof. 17, with the vi and wi ranging over the standard basis {em,n } of l2 (Z2 ). We will show that [Vsr ]1 is eventually within the -neighborhood of [Vs ]1 for the seminorm ||| · |||; the corresponding assertion with r and switched is proven similarly.

Let V and W be quantum pseudometrics on von Neumann algebras M ⊆ B(H) and N ⊆ B(K). (a) The metric product V⊗F W is the meet of the quantum pseudometrics V⊗W0 = {Vt ⊗Wt0 } and V0 ⊗W = {Vt0 ⊗Wt } where V0 = {Vt0 } and W0 = {Wt0 } are the trivial quantum pseudometrics with Vt0 = B(H) and Wt0 = B(K) for all t. (b) The natural embeddings ι1 : A → A ⊗ IK and ι2 : B → IH ⊗ B of M and N into M⊗N realize M and N as metric subobjects of the metric product. (c) V⊗F W is a quantum metric if and only if both V and W are quantum metrics.

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 . Conversely, let A ∈ B(H) − V0 . 7 we can then ﬁnd projections P, Q ∈ M⊗B(l2 ) such that P (B ⊗ I)Q = 0 for all B ∈ Vt but P (A ⊗ I)Q = 0. It follows that P is orthogonal to Q, and thus the range of Q cannot be invariant for A ⊗ I. So A ⊗ I does not commute with Q, and hence A ⊗ I ∈ N . This completes the proof that N = V0 ⊗ I.