The ideal structure of Toeplitz algebras
The authors study the algebra \({\mathcal T}(\Gamma)\subset B(\ell^2(\Gamma^+)\) generated by isometries of a totally ordered Abelian group \(\Gamma\), defined on the standard basis as shifts \(Z_xe_y=e_{x+y}\), \(x\in\Gamma^+\). The structure of ideals is well known when either \(\Gamma\subset\mathbb{R}\) or \(\Gamma\) is finitely generated. The problem of describing the ideal structure for a non-finitely generated totally ordered group is considered. A parametrization by \(X\) of the set of primitive ideals of the totally ordered set \(\Sigma(\Gamma)\) of ordered ideals \({\mathcal T}(\Gamma)\) is described. Also the hull-kernel topology on \(X\) is identified when the chain of order ideals in \(\Gamma\) is isomorphic to a subset of \(\{-\infty\}\cup\mathbb{Z}\cup\{\infty\}\).
scientific article; zbMATH DE number 2101681