The solvable structure of the \(C^\ast\)-algebras of certain successive semi-direct products (Q1772545)

The paper is closely linked to the author's numerous works on the structure of group \(C^*\)-algebras \(C^*(G)\) for certain Lie groups \(G\). It shows, in the case of (duly adjusted) successive semi-direct products of \({\mathbb C}^n\) by \(\mathbb R\), \(\mathbb T\) or \(\mathbb Z\), that \(C^*(G)\) has finite composition series of ideals whose quotients are tensor products involving abelian \(C^*\)-algebras, noncommutative tori and/or all compact operators over an \(H\)-space. Some other groups \(G\) are analyzed as well. As a corollary the estimates of stable and connected stable ranks of these \(C^*(G)\) are obtained in terms of \(G\). Also the author conjectures that the introduced class of \(C^*\)-solvable \(C^*\)-algebras (cf. the notion of solvability in [\textit{C. Schochet}, Pac. J. Math. 98, 443-458 (1982; Zbl 0439.46043)]) contains \(C^*(G)\) for any solvable Lie group.