If G is a discrete group, the algebra CD(G) of convolution dominated operators on l²(G) (see Definition 1 below) is canonically isomorphic to a twisted L¹-algebra . 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 l²(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].
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-bc78-0-10,
author = {Gero Fendle and Karlheinz Gr\"ochenig and Michael Leinert},
title = {On spectrality of the algebra of convolution dominated operators},
journal = {Banach Center Publications},
volume = {75},
year = {2007},
pages = {145-149},
zbl = {1131.47061},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc78-0-10}
}
Gero Fendle; Karlheinz Gröchenig; Michael Leinert. On spectrality of the algebra of convolution dominated operators. Banach Center Publications, Tome 75 (2007) pp. 145-149. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc78-0-10/