Advances in Operator Theory

Convolution dominated operators on compact extensions of abelian groups

Gero Fendler and Michael Leinert

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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
Received: 16 December 2017
Accepted: 3 April 2018
First available in Project Euclid: 27 April 2018

Permanent link to this document
https://projecteuclid.org/euclid.aot/1524816020

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

Mathematical Reviews number (MathSciNet)
MR3867336

Zentralblatt MATH identifier
06946445

Subjects
Primary: 47B35: Toeplitz operators, Hankel operators, Wiener-Hopf operators [See also 45P05, 47G10 for other integral operators; see also 32A25, 32M15]
Secondary: 43A20: $L^1$-algebras on groups, semigroups, etc.

Keywords
convolution dominated operators generalised $L^1$-algebras symmetric locally compact group

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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