Convolution dominated operators on compact extensions of abelian groups

2017/11/23 by Fendler, Gero, Leinert, Michael
#43A20 (Secondary) #47B35 (Primary) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1711.08638

Abstract

If G is a locally compact group, CD(G) the algebra of convolution dominated operators on L²(G) then an important question is: Is ℂ1+CD(G) (respectively CD(G) if G is discrete) inverse-closed in the bounded operators on L²(G)? In this note we answer this question in the affirmative provided G is such that one of the following properties is fulfilled (1) There is a discrete, rigidly symmetric, and amenable subgroup H⊂ G and a (measurable) relatively compact neighbourhood of the identity U invariant under conjugation by elements of H such that \hU : h∈ H\ is a partition of G. (2) The commutator subgroup of G is relatively compact. (If G is connected this just means that G is an IN group.) All known examples where CD(G) is inverse-closed in B(L²(G)) are covered by this.

Related

• On Convolution Dominated Operators
• Convolutions on compact groups and Fourier algebras of coset spaces
• Spectral analysis for convolution operators on locally compact groups
• Discrete convolution operators and Riesz systems generated by actions of\n abelian groups
• Band-dominated and Fourier-band-dominated operators on locally compact abelian groups