On spectrality of the algebra of convolution dominated operators
Tom 78 / 2007
Banach Center Publications 78 (2007), 145-149
MSC: Primary 47B35; Secondary 43A20.
DOI: 10.4064/bc78-0-10
Streszczenie
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].
