On spectrality of the algebra of convolution dominated operators

Volume 78 / 2007

Gero Fendle, Karlheinz Gröchenig, Michael Leinert Banach Center Publications 78 (2007), 145-149 MSC: Primary 47B35; Secondary 43A20. DOI: 10.4064/bc78-0-10


If $G$ is a discrete group, the algebra $CD(G) $ of convolution dominated operators on $l^{2}(G)$ (see Definition 1 below) is canonically isomorphic to a twisted $L^{1}$-algebra $l^{1} (G, l^{\infty}(G),T)$. For amenable and rigidly symmetric $G$ we use this to show that any element of this algebra is invertible in the algebra itself if and only if it is invertible as a bounded operator on $l^2(G)$, i.e. $CD(G)$ is spectral in the algebra of all bounded operators. For $G$ commutative, this result is known (see [1], [6]), for $G$ noncommutative discrete it appears to be new. This note is about work in progress. Complete details and more will be given in [3].


  • Gero FendleFinstertal 16
    D-69514 Laudenbach, Germany
  • Karlheinz GröchenigFakultät für Mathematik
    Universität Wien
    Nordbergstrasse 15
    A-1090 Wien, Austria
  • Michael LeinertInstitut für Angewandte Mathematik
    Fakultät für Mathematik
    Im Neuenheimer Feld 288
    D-69120 Heidelberg, Germany

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