Convolution dominated operators on compact extensions of abelian groups

Abstract

If $G$ is a locally compact group, $CD(G)$ the algebra of convolution dominated operators on $L^2(G)$, then an important question is: Is $\mathbb {C}1+CD(G)$ (or $CD(G)$ if $G$ is discrete) inverse-closed in the algebra of bounded operators on $L^2(G)$?

In this note we answer this question in the affirmative, provided $G$ is such that one of the following properties is satisfied.

1. There is a discrete, rigidly symmetric, and amenable subgroup $H\subset G$ and a (measurable) relatively compact neighbourhood of the identity $U$, invariant under conjugation by elements of $H$, such that $\{hU\;:\;h\in H\}$ 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 $B(L^2(G))$ are covered by this.

Article information

Source
Adv. Oper. Theory, Volume 4, Number 1 (2019), 99-112.

Dates
Accepted: 3 April 2018
First available in Project Euclid: 27 April 2018

https://projecteuclid.org/euclid.aot/1524816020

Digital Object Identifier
doi:10.15352/aot.1712-1275

Mathematical Reviews number (MathSciNet)
MR3867336

Zentralblatt MATH identifier
06946445

Citation

Fendler, Gero; Leinert, Michael. Convolution dominated operators on compact extensions of abelian groups. Adv. Oper. Theory 4 (2019), no. 1, 99--112. doi:10.15352/aot.1712-1275. https://projecteuclid.org/euclid.aot/1524816020

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