By Greg Kuperberg

ISBN-10: 0821853414

ISBN-13: 9780821853412

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

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

**Example text**

5. Quantum tori We formulate a notion of translation invariant quantum pseudometrics on quantum tori. 7 of [35], it is then straightforward to deduce strong structural information about translation invariant quantum pseudometrics. Quantum tori are the simplest examples of noncommutative manifolds. They are related to the quantum plane, which plays the role of the phase space of a spinless one-dimensional particle. The classical version of such a system has phase space R2 , with the point (q, p) ∈ R2 representing a state with position q and momentum p, so that the position and momentum observables are just the coordinate functions on phase space.

5) which states that every such map can be expressed as an inﬂation followed by a restriction followed by an isomorphism. Since this expression is not unique, if we deﬁned L(φ) in the concrete way indicated below then the deﬁnition would appear to be ambiguous. But the fact that this deﬁnition is equivalent to the intrinsic one given above means that there is actually no real ambiguity. 31. Let V and W be quantum pseudometrics on von Neumann algebras M ⊆ B(H) and N ⊆ B(K) and let φ : M → N be a unital weak* continu˜ be a Hilbert space, R a projection in B(K)⊗M ˜ , and ous ∗-homomorphism.

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.

### A von Neumann algebra approach to quantum metrics. Quantum relations by Greg Kuperberg

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