Quantum Differentiability on Noncommutative Euclidean Spaces

Abstract

We study the topic of quantum differentiability on quantum Euclidean d-dimensional spaces (otherwise known as Moyal d-spaces), and we find conditions that are necessary and sufficient for the singular values of the quantised differentialdx to have decay O(n −α ) for 0 < α ≤ 1 d . This result is substantially more difficult than the analogous problems for Euclidean space and for quantum d-tori.2000 Mathematics Subject Classification: Primary: 46G05. Secondary: 47L10, 58B34.

The Noncommutative Geometry of the Landau Hamiltonian: Metric Aspects

“…The latter result provides a direct generalization of the first Connes' formula obtained in [6] for the discrete case. Even more related to Theorem 1.3 is the work [54]. In the latter paper a formula of the type (1.6) is derived for the noncommutative Euclidean space (or Moyal plane) which is indeed similar to the algebra C B introduced above.…”

“…In the latter paper a formula of the type (1.6) is derived for the noncommutative Euclidean space (or Moyal plane) which is indeed similar to the algebra C B introduced above. The main difference between the result in [54] and Theorem 1.3 lies in the choice of the Dirac operator. In fact, in [54] the authors deal with the operator D (described above) which has a non-compact resolvent, while formula (1.6) is obtained from the Dirac operator D B which has compact resolvent.…”

“…In particular, the Dirac operator D allows a complete description of the QHE and (other topological phases) in the continuum [9]. A closely related approach, already considered in the literature in different contexts [27,43,54], consists in considering the Dirac operator D = 2 j=1 (− i ∂ j ) ⊗ σ j which is the Fourier transform of D. The latter choice also allows to prove a formula of the type (1.2) by combining the result of [4] with the observation that D 2 is proportional to the Laplacian ∆ on R 2 . A completely different approach consists in constructing a compact spectral triple for the magnetic algebra C B .…”

Endpoint weak Schatten class estimates and trace formula for commutators of Riesz transforms with multipliers on Heisenberg groups

“…In the words of Connes, by such a formula one can "pass from quantized 1-forms to ordinary forms, not by a classical limit, but by a direct application of the Dixmier trace". Extensions and analogies of this formula [14,Theorem 3(iii)] in various non-compact and noncommutative settings have been obtained more recently, see for example the results [32], [36] and [35] on the non-compact manifold R d , the quantum Euclidean space and the quantum tori, respectively.…”

“…We are uncertain at this stage whether our trace formula (1.2) is relevant to noncommutative geometry, however there are direct applications in harmonic analysis. For example, following from [36,Corollary 1.5], one can apply the trace formula (1.2) to give an alternative proof of [22, (2) in Theorem 1.1].…”