Strobl22

Applied Harmonic Analysis and Friends

June 19th - 25th 2022

Strobl, AUSTRIA

"Bundle Structures on Feichtinger’s Algebras"

Lamando, Arvin

Let $G$ be a locally compact abelian group, $\Lambda$ a closed subgroup of $G\times \widehat{G}$ with $2$-cocycle $c: \Lambda \times \Lambda \to \mathbb{C}.$ Given the dual $\Lambda^{\circ}$ of $\Lambda,$ we have the following Feichtinger algebras $S_0(\Lambda,c)$, $S_0(\Lambda^{\circ},\overline{c})$, and $S_0(G\times \widehat{G})$ as dense subspaces of $C^*(\Lambda,c),$ $C^*(\Lambda^{\circ},\overline{c}),$ and $\mathcal{E}(G)$ respectively. The construction of the so-called Heisenberg bimodule $\mathcal{E}(G)$ is such that it is an imprimitivity equivalence bimodule between $C^*$-algebras $C^*(\Lambda,c)$ and $C^*(\Lambda^{*},\overline{c}).$ The Heisenberg bimodule has been proposed to be a candidate extension-function-space for time-frequency analysis as it preserves certain properties enjoyed by the Feichtinger's algebra like the fundamental identity of Gabor analysis and the boundedness of the associated analysis and synthesis operators of the Gabor systems with windows coming from $\mathcal{E}(G)$. We plan to further study this line of $C^*$-algebraic approach by looking at how (possibly twisted) partial actions on the Feichtinger's algebra induce bundles of Heisenberg bimodules, which further induce cross-sectional algebras that are of interest for studying continuously varying time-frequency systems.
http://univie.ac.at/projektservice-mathematik/e/talks/Lamando_2022-02_Abstract__STROBL2022.pdf

« back