By Greg Kuperberg

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

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

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A metric subobject of M is a unital von Neumann subalgebra N ⊆ M together with the quantum pseudometric VN = {W : V ≤ W and N ⊆ W0 } where W ranges over W*-ﬁltrations of B(H). In other words, VN is the meet of all quantum pseudometrics on N that dominate V. 39. Let V be a quantum pseudometric on a von Neumann algebra M ⊆ B(H) and let N ⊆ M be a metric subobject of M. (a) The inclusion map ι : N → M is a co-contraction morphism (equipping N with the quantum pseudometric VN ). (b) If W is any quantum pseudometric on N which makes the inclusion map a co-contraction morphism then the identity map from N to itself is a co-contraction morphism from the pseudometric W to the pseudometric VN .

The most useful generalization of the normal spatial tensor product of von Neumann algebras to dual operator spaces is the normal Fubini tensor product [12]. , the weak* closure of the algebraic tensor product). Abstractly, V⊗F W is characterized as the dual of the projective tensor product of the preduals of V and W [2, 25]: ˆ ∗ )∗ . V⊗F W ∼ = (V∗ ⊗W The normal spatial tensor product is always contained in the normal Fubini tensor product but this inclusion may be strict. Thus, the equality V⊗F W = (V⊗B(K)) ∩ (B(H)⊗W), which is crucial for the following proof, fails in general for the normal spatial tensor product.

Then the inner products U− k l to the inner products U−r V−r wi , vi as → r, uniformly in k and l since these inner products are zero for suﬃciently large k and l. Next let > 0 and observe that the inner products σN (A)wi , vi converge to the inner products Awi , vi as N → ∞, uniformly in A ∈ [Vsr ]1 . 35). By the preceding observation, if k k l ak,l U− V−l ∈ Vs where σN (A) = ak,l U−r V−r then we will we write σN (A) = have |||σN (A) − A||| ≤ |||σN (A) − σN (A)||| + |||σN (A) − A||| with the ﬁrst term on the right going to zero as → r and the second going to zero as N → ∞, both uniformly in A ∈ [Vsr ]1 .