Quantum Differentiability on Quantum Tori

Abstract

We provide a full characterisation of quantum differentiability (in the sense of Connes) on quantum tori. We also prove a quantum integration formula which differs substantially from the commutative case.2000 Mathematics Subject Classification: Primary: 46G05. Secondary: 47L10, 58B34.

Quantum Differentiability on Noncommutative Euclidean Spaces

“…. The proof given here is similar to the corresponding result on quantum tori [45], relying heavily on the Cwikel type estimate stated in the last section.…”

“…This section is devoted to the proof of Theorem 1.6, which is an essential ingredient for our proof of Theorem 1.2 i.e., the computation of ϕ(|dx| d ) when x ∈ L ∞ (R d θ ) ∩Ẇ 1 d (R d θ ) and ϕ is a continuous normalised trace on L 1,∞ . One powerful tool used in [45] for quantum tori is the theory of noncommutative pseudodifferential operators. The proof in [45] proceeds by viewing the quantised differentialdx = i[sgn(D), 1⊗x] as a pseudodifferential operator, then determining its (principal) symbol and order, and finally appealing to Connes' trace formula as obtained in [46].…”

“…Recently the authors have established a characterisation of the L d,∞ -ideal membership of quantised differentials for noncommutative tori [45]. The primary result of [45] is as follows. Let θ be an antisymmetric real d × d matrix with d > 2, and consider the noncommutative tori T d θ .…”

“…An element x ∈ L 2 (T d θ ) belongs to the (noncommutative) homogeneous Sobolev spaceẆ 1 d (T d θ ) if and only if its quantised differentialdx has bounded extension in L d,∞ . The quantum torus analogue of (1.2) is also obtained as [45,Theorem 1.2]: for x ∈Ẇ 1 d (T d θ ), there is a certain constant c d such that for any continuous normalised trace ϕ on L 1,∞ we have…”

The Noncommutative Geometry of the Landau Hamiltonian: Metric Aspects

“…It is important to point out that formulas of the type (1.6) have already been investigated in the literature in contexts not necessarily related to the QHE. For instance, in [53] a similar formula has been derived for the case of a (d-dimensional) noncommutative torus. The latter result provides a direct generalization of the first Connes' formula obtained in [6] for the discrete case.…”