Noncommutative solenoids and their projective modules

Let p be prime. A noncommutative p-solenoid is the C*-algebra of Z[1/p] x Z[1/p] twisted by a multiplier of that group, where Z[1/p] is the additive subgroup of the field Q of rational numbers whose denominators are powers of p. In this paper, we survey our classification of these C*-algebras up to *-isomorphism in terms of the multipliers on Z[1/p], using techniques from noncommutative topology. Our work relies in part on writing these C*-algebras as direct limits of rotation algebras, i.e. twisted group C*-algebras of the group Z^2 thereby providing a mean for computing the K-theory of the noncommutative solenoids, as well as the range of the trace on the K_0 groups. We also establish a necessary and sufficient condition for the simplicity of the noncommutative solenoids. Then, using the computation of the trace on K_0, we discuss two different ways of constructing projective modules over the noncommutative solenoids.Comment: To appear in the AMS Contemporary Mathematics volume entitled Commutative and Noncommutative Harmonic Analysis and Applications edited by Azita Mayeli, Alex Iosevich, Palle E. T. Jorgensen and Gestur Olafsson. 19 Page

Interpolation in Wavelet Spaces and the HRT-Conjecture

We investigate the wavelet spaces $\mathcal{W}_{g}(\mathcal{H}_{\pi})\subset L^{2}(G)$ arising from square integrable representations $\pi:G \to \mathcal{U}(\mathcal{H}_{\pi})$ of a locally compact group $G$. We show that the wavelet spaces are rigid in the sense that non-trivial intersection between them imposes strong conditions. Moreover, we use this to derive consequences for wavelet transforms related to convexity and functions of positive type. Motivated by the reproducing kernel Hilbert space structure of wavelet spaces we examine an interpolation problem. In the setting of time-frequency analysis, this problem turns out to be equivalent to the HRT-Conjecture. Finally, we consider the problem of whether all the wavelet spaces $\mathcal{W}_{g}(\mathcal{H}_{\pi})$ of a locally compact group $G$ collectively exhaust the ambient space $L^{2}(G)$. We show that the answer is affirmative for compact groups, while negative for the reduced Heisenberg group.Comment: Added a relevant citation and made minor modifications to the expositio

Square-integrability of multivariate metaplectic wave-packet representations

This paper presents a systematic study for harmonic analysis of metaplectic wave-packet representations on the Hilbert function space L2(Rd). The abstract notions of symplectic wave-packet groups and metaplectic wave-packet representations will be introduced. We then present an admissibility condition on closed subgroups of the real symplectic group Sp(Rd), which guarantees the square-integrability of the associated metaplectic wave-packet representation on L2(Rd)