Continuity of Spectra in Rieffel’s Pseudodifferential Calculus

Covariant Fields of C*-Algebras under Rieffel Deformation

“…That paper has been submitted after the completion of the present article and refers to it explicitly. In [1] the proof relies on a recently found connection between Rieffel deformation and twisted crossed products [2] and on the deep analysis from [20], therefore being less direct or selfcontained than the present one.…”

“…Using this notion, Theorem 1.1 is now fully justified. Another proof, based on the theory of twisted crossed products and on results from the deep work [20] can be found in [1], as we indicated in the introduction.…”

“…A for any f ∈ A ∞ = A ∞ . Replacing here f by δ α f for every multi-index α, it follows that on A ∞ the topology given by the semi-norms (1) is finer than the one given by the semi-norms (2). As a consequence of Theorem 7.5 in [21], the role of the C * -algebras A and A can be reversed: one obtains A as the deformation of A by replacing the skew-symmetric form…”

“…Moreover, for us this was the main technical problem in proving Theorem 4.3; we solved it using the highly nontrivial Dixmier-Malliavin theorem [5]. Our detailed proof based on the more traditional approach is intended to support future work on spectral theory of pseudodifferential operators (see also [1,14]), as well as applications to quantum mechanics. The lower-semi-continuity part (contained in Section 4), often described as difficult, is not treated in [9,10] and seems not to have correspondence in the literature.…”

“…Let us mention that a different proof of the main result (Theorem 1.1) has been given in [1], where this result is needed as an important ingredient in proving spectral continuity properties of quantum Hamiltonians defined as phase-space anisotropic pseudodifferential operators. That paper has been submitted after the completion of the present article and refers to it explicitly.…”