By J. Hirschfeld, W.H. Wheeler

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**Example text**

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 , . . , wn ∈ H the sets [Vs ]1 and [Vsλ ]1 are, respectively, eventually within the -neighborhoods of the sets [Vtλ ]1 and [Vt ]1 for the seminorm |||A||| = | Awi , vi |.

Finally, we note that the quotient, subobject, and product constructions discussed above reduce to the standard notions in the atomic abelian case. 43. 5). 5) is a metric quotient of M. Every metric quotient of M is of this form. 2), the von Neumann algebra ˜ = {Mf : f ∈ l∞ (X), x ∼ y ⇒ f (x) = f (y)} ⊆ M M equipped with the quantum pseudometric Vd˜ is a metric subobject of M. Every metric subobject of M is of this form. (c) The metric product of M and N is the von Neumann algebra M⊗N ∼ = l∞ (X × Y ) equipped with the quantum pseudometric VdX×Y associated to the pseudometric dX×Y ((x1 , y1 ), (x2 , y2 )) = max{d(x1 , x2 ), d (y1 , y2 )}.

The quantum torus von Neumann algebra for the given value of mann algebra W ∗ (U , V ) generated by U and V . is the von Neu- 44 3. 38). If is an irrational multiple of π then W ∗ (U , V ) is a hyperﬁnite II1 factor. We will not need this fact. Conjugating U and V by the Fourier transform F : L2 (T2 ) → l2 (Z2 ) yields the operators ˆ f (x, y) U = eix f x, y − Vˆ f (x, y) = eiy f x + ,y 2 2 ˆ , Vˆ ) reducing to the algebra of bounded multiplication on L2 (T2 ), with W ∗ (U operators when = 0. However, for our purposes the l2 (Z2 ) picture is more convenient.