Advances in Operator Theory

Convolution dominated operators on compact extensions of abelian groups

Gero Fendler and Michael Leinert

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

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

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

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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.

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


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.

Export citation


  • A. G. Baskakov, Abstract harmonic analysis and asymptotic estimates for elements of inverse matrices, (Russian); translated from Mat. Zametki 52 (1992), no. 2, 17–26, 155 Math. Notes 52 (1992), no. 1-2, 764–771 (1993).
  • A. G. Baskakov, Asymptotic estimates for elements of matrices of inverse operators, and harmonic analysis, (Russian); translated from Sibirsk. Mat. Zh. 38 (1997), no. 1, 14–28, i Siberian Math. J. 38 (1997), no. 1, 10–22.
  • A. G. Baskakov, Estimates for the elements of inverse matrices, and the spectral analysis of linear operators, (Russian); translated from Izv. Ross. Akad. Nauk Ser. Mat. 61 (1997), no. 6, 3–26 Izv. Math. 61 (1997), no. 6, 1113–1135.
  • I. Beltiţă and D. Beltiţă, Erratum to: Inverse-closed algebras of integral operators on locally compact groups, Ann. Henri Poincaré 16 (2015), no. 5, 1307–1309.
  • S. Bochner and R. S. Phillips, Absolutely convergent Fourier expansions for non-commutative normed rings, Ann. of Math. (2), 43 (1942), 409–418.
  • B. Farrell and T. Strohmer, Inverse-closedness of a Banach algebra of integral operators on the Heisenberg group, J. Operator Theory 64 (2010), no. 1, 189–205.
  • G. Fendler, K. Gr öchenig, and M. Leinert, Convolution-dominated operators on discrete groups, Integral Equations Operator Theory 61 (2008), no. 4, 493–509.
  • G. Fendler, K. Gr öchenig, and M. Leinert, Convolution-dominated integral operators, Noncommutative harmonic analysis with applications to probability II, 121–127, Banach Center Publ., 89, Polish Acad. Sci. Inst. Math., Warsaw, 2010.
  • G. Fendler and M. Leinert, On convolution dominated operators, Integral Equations Operator Theory 86 (2016), no. 2, 209–230.
  • I. Gohberg, M. A. Kaashoek, and H. J. Woerdeman, The band method for positive and contractive extension problems, J. Operator Theory 22 (1989), no. 1, 109–155.
  • F. P. Greenleaf, Invariant means on topological groups and their applications, Van Nostrand Mathematical Studies, No. 16, Van Nostrand Reinhold Co., New York-Toronto, Ont.-London, 1969.
  • K. Gr öchenig and M. Leinert, Wiener's lemma for twisted convolution and Gabor frames, J. Amer. Math. Soc. 17 (2004), 1–18.
  • A. Hulanicki, On the spectrum of convolution operators on groups with polynomial growth, Invent. Math. 17 (1972), 135–142.
  • K. Iwasawa, Topological groups with invariant compact neighborhoods of the identity, Ann. of Math. (2) 54 (1951), 345–348.
  • V. G. Kurbatov, Algebras of difference and integral operators, Funktsional. Anal. i Prilozhen. 24 (1990), no. 2, 87–88.
  • V. G. Kurbatov, Functional differential operators and equations, Mathematics and its Applications, 473. Kluwer Academic Publishers, Dordrecht, 1999.
  • V. G. Kurbatov, Some algebras of operators majorized by a convolution, International Conference on Differential and Functional Differential Equations (Moscow, 1999). Funct. Differ. Equ. 8 (2001), no. 3-4, 323–333.
  • H. Leptin, Verallgemeinerte $L^{1}$-Algebren, Math. Ann. 159 (1965), 51–76.
  • H. Leptin, Verallgemeinerte ${L}^{1}$-Algebren und projektive Darstellungen lokal kompakter Gruppen, I, Invent. Math. 3 (1967), 257–281.
  • H. Leptin, Darstellungen verallgemeinerter $L^1$-Algebren, (German) Invent. Math. 5 (1968), 192–215.
  • H. Leptin and D. Poguntke, Symmetry and non-symmetry for locally compact groups, J. Funct. anal. 33 (1979), 119–134.
  • Q. Sun, Wiener's lemma for infinite matrices with polynomial off-diagonal decay, C. R. Math. Acad. Sci. Paris 340 (2005), no. 8, 567–570.
  • N. Wienerm, Tauberian theorems, Ann. of Math. (2) 33 (1932), no. 1, 1–100.