On the classification of commutator ideals
Let \(\Gamma\) be a subgroup of \({\mathbb{R}}\), and let \(\Gamma_+=\Gamma \cap [0,\infty)\). Then the \(C^*\)-algebra generated by isometric representations of \(\Gamma_+\) is known (by work of R. Douglas) to have only one isomorphism class, represented \({\mathcal I}(\Gamma)\). The commutator ideal in \({\mathcal I}(\Gamma)\) is denoted \({\mathcal C}(\Gamma)\). It is simple. Since \({\mathbb{R}}\) is selfdual, a given discrete subgroup \(\Gamma\) of \({\mathbb{R}}\) yields an embedding of \({\mathbb{R}}\) in \({\hat \Gamma}\) inducing the foliation \(C^*\)-algebra \(C_ 0({\mathbb{R}})\rtimes \Gamma\) contained in \(\hat {\mathcal C}(\Gamma)\) a certain AF-crossed product algebra. The foliation algebra is viewed as the smooth part of \(\hat {\mathcal C}(\Gamma)\). The normalized trace \(\tau_{\Gamma}\) induces a homomorphism \((\tau_{\Gamma})_*\) on K-theory (with values in \({\mathbb{R}})\), and it is proved that the range of \((\tau_{\Gamma})_*\) on \(K_ 0\) is \(\Gamma\). This holds for \(K_ 0\) of both \({\mathcal C}(\Gamma)\) and \(\hat {\mathcal C}(\Gamma)\). The corresponding complete isomorphism invariant is identified relative to (i) order isomorphism on the group level (i.e., the discrete subgroups \(\Gamma)\), (ii) \(C^*\)-isomorphism of the corresponding \({\mathcal I}(\Gamma)'s,\) (iii) strong Morita equivalence of the \({\mathcal I}(\Gamma)'s,\) (iv) isomorphism of the respective \({\mathcal C}(\Gamma)'s.\) For a pair of groups \(\Gamma\) and \(\Gamma\) ', isomorphism is specified by a \(c>0\) such that \(c\Gamma =\Gamma '\). This is used to compute the K- theory associated to a pair \(\Gamma_ 0\) and \(\Gamma\) where \(\Gamma_ 0\) is a subgroup of \(\Gamma\) and further specifics for groups \(\Gamma\) which are assumed to contain a pair of elements linearly independent over \({\mathbb{Q}}\). Further classification results for the commutator ideals are obtained.
scientific article; zbMATH DE number 4065715
scientific article