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Winter 2019 Convolution dominated operators on compact extensions of abelian groups
Gero Fendler, Michael Leinert
Adv. Oper. Theory 4(1): 99-112 (Winter 2019). DOI: 10.15352/aot.1712-1275

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.

Citation

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Gero Fendler. Michael Leinert. "Convolution dominated operators on compact extensions of abelian groups." Adv. Oper. Theory 4 (1) 99 - 112, Winter 2019. https://doi.org/10.15352/aot.1712-1275

Information

Received: 16 December 2017; Accepted: 3 April 2018; Published: Winter 2019
First available in Project Euclid: 27 April 2018

zbMATH: 06946445
MathSciNet: MR3867336
Digital Object Identifier: 10.15352/aot.1712-1275

Subjects:
Primary: 47B35
Secondary: 43A20

Rights: Copyright © 2019 Tusi Mathematical Research Group

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Vol.4 • No. 1 • Winter 2019
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