\(K\)-theory of the full \(C^*\)-algebra of \(SL_{3}(\mathbb C)\)
Let \(G\) be \(SL_3({\mathbb C})\) and let \(\lambda:C^*(G)\to C^*_r(G)\) and \(\varepsilon:C^*(G)\to{\mathbb C}\) be the homomorphisms induced by the regular representation and by the trivial representation respectively. It is proved that \(\lambda\oplus\varepsilon\) induces a \(KK\)-equivalence between \(C^*(G)\) and \(C^*_r(G)\oplus{\mathbb C}\). This result together with calculation of the \(K\)-groups for \(C^*_r(G)\) made by \textit{A. Wassermann} in [C. R. Acad. Sci., Paris, Sér. I 304, 559-562 (1987; Zbl 0615.22011)] shows that \(K_0(C^*(G))={\mathbb Z}^\infty\) and \(K_1(C^*(G))=0\). Let \(P\subset G\) be a maximal parabolic subgroup and let \(j_P\rtimes G:C^*(G)\to C(G/P)\rtimes G\) denote the homomorphism induced by the unital \(G\)-equivariant homomorphism \(j_P:{\mathbb C}\to C(G/P)\). Let also \(\lambda^{C(G/P)}:C(G/P)\rtimes G\to C(G/P)\rtimes_r G\) be the homomorphism from the full crossed product to the reduced one. Let \(I=\text{Ker}(\lambda)\) and \(I^{C(G/P)}=\text{Ker}(\lambda^{C(G/P)})\). It is proved that \(j_P\rtimes G\oplus\varepsilon\) induces an isomorphism between \(I\) and \(I^{C(G/P)}\oplus{\mathbb C}\) and that \(I^{C(G/P)}\) is contractible.
scientific article; zbMATH DE number 1836678
scientific article