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 Gc is C∗-unique and symmetric, then the twisted convolution algebra L1(G, c) is spectrally invariant in B(H) for any faithful ∗-representation of L1(G, c) as bounded operators on a Hilbert space H. As an application of this result we give a proof of the statement that if is a closed cocompact subgroup of the phase space of a locally compact abelian group G', and if g is some function in the Feichtinger algebra S0(G') that generates a Gabor frame for L2(G') over , then both the canonical dual atom and the canonical tight atom associated to g are also in S0(G'). We do this without the use of periodization techniques from Gabor analysis.