Heisenberg modules as function spaces
| Autor: | Ulrik Enstad, Are Austad |
|---|---|
| Rok vydání: | 2020 |
| Předmět: |
Group (mathematics)
Function space Applied Mathematics General Mathematics Lattice (group) Hilbert space Mathematics - Operator Algebras Second-countable space 42C15 46L08 43A70 Functional Analysis (math.FA) Combinatorics Mathematics - Functional Analysis symbols.namesake Bounded function FOS: Mathematics symbols Locally compact space Abelian group Operator Algebras (math.OA) Analysis Mathematics |
| ISSN: | 1069-5869 |
| Popis: | Let $\Delta$ be a closed, cocompact subgroup of $G \times \widehat{G}$, where $G$ is a second countable, locally compact abelian group. Using localization of Hilbert $C^*$-modules, we show that the Heisenberg module $\mathcal{E}_{\Delta}(G)$ over the twisted group $C^*$-algebra $C^*(\Delta,c)$ due to Rieffel can be continuously and densely embedded into the Hilbert space $L^2(G)$. This allows us to characterize a finite set of generators for $\mathcal{E}_{\Delta}(G)$ as exactly the generators of multi-window (continuous) Gabor frames over $\Delta$, a result which was previously known only for a dense subspace of $\mathcal{E}_{\Delta}(G)$. We show that $\mathcal{E}_{\Delta}(G)$ as a function space satisfies two properties that make it eligible for time-frequency analysis: Its elements satisfy the fundamental identity of Gabor analysis if $\Delta$ is a lattice, and their associated frame operators corresponding to $\Delta$ are bounded. Comment: 24 pages; several changes have been made to the presentation, while the content remains essentially unchanged; to appear in Journal of Fourier Analysis and Applications |
| Databáze: | OpenAIRE |
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