Spectral Invariance of $$*$$-Representations of Twisted Convolution Algebras with Applications in Gabor Analysis

Abstract

We show spectral invariance for faithful $$*$$ ∗ -representations for a class of twisted convolution algebras. More precisely, if G is a locally compact group with a continuous 2-cocycle c for which the corresponding Mackey group $$G_c$$ G c is $$C^*$$ C ∗ -unique and symmetric, then the twisted convolution algebra $$L^1 (G,c)$$ L 1 ( G , c ) is spectrally invariant in $${\mathbb {B}}({\mathcal {H}})$$ B ( H ) for any faithful $$*$$ ∗ -representation of $$L^1 (G,c)$$ L 1 ( G , c ) as bounded operators on a Hilbert space $${\mathcal {H}}$$ H . As an application of this result we give a proof of the statement that if $$\Delta $$ Δ is a closed cocompact subgroup of the phase space of a locally compact abelian group $$G'$$ G ′ , and if g is some function in the Feichtinger algebra $$S_0 (G')$$ S 0 ( G ′ ) that generates a Gabor frame for $$L^2 (G')$$ L 2 ( G ′ ) over $$\Delta $$ Δ , then both the canonical dual atom and the canonical tight atom associated to g are also in $$S_0 (G')$$ S 0 ( G ′ ) . We do this without the use of periodization techniques from Gabor analysis.

Deformations and Balian–Low theorems for Gabor frames on the adeles

“…With all the previous lemmas and propositions taken care of, the proof below is exactly the same as the proof of [16,Theorem 3.8]. The deep fact that a Gabor frame G(g, Δ) with g ∈ S 0 (G) admits a dual frame with window also in S 0 (G) was proved for G = R n in [22] and generalized to locally compact abelian groups in [3]. Theorem 4.6.…”

“…Theorem 4.6. Let G be a locally compact abelian group, and let Δ be a lattice in G × G. Then the set [3,Theorem B] we can find a dual frame G(h 0 , α 0 (Δ)) with h 0 ∈ S 0 (G). Then S g 0 ,h 0 ,α 0 (Δ) = I.…”

“…Our main tools are the duality theory for uniform Gabor frames on LCA groups [31], continuity of various maps between the Feichtinger algebra of different groups [28], and the existence of dual windows in the Feichtinger algebra [3,22]. At the heart of our approach is the continuity of dilation on S 0 (G) by automorphisms from Aut(G), see Theorem 4.3.…”