Complex group rings and group \(\mathrm{C}^*\)-algebras of group extensions (Q6063392)

In the paper, the authors study the Kaplansky's conjectures for complex group rings and group C*-algebras of group extensions. Recall that the Kaplansky's complex zero-divisor (idempotent) conjecture asserts that the group ring \(\mathbb {C}[G]\) of a torsion-free group \(G\) has no zero-divisors (non-trivial idempotents). The Kadison-Kaplansky conjecture asserts that the reduced group C*-algebra of a (discrete) torsion-free group \(G\) has no non-trivial idempotents. A group \(H\) is a unique product group if for any two non-empty finite subsets \(A, B \subseteq H\) there exists at least one element \(h \in H\) which has a unique representation of the form \(h = ab\) with \(a\in A\) and \(b \in B\). In particular, the authors proved that if \(G\) is an extension of a unique product group \(H\) be a group \(N \) and the group \(N\) satisfies the Kaplansky's complex zero-divisor conjecture then \(G\) satisfies Kaplansky's complex zero-divisor conjecture and the complex idempotent conjecture (Corollary 3.6). The authors also proved that if \(G\) is a central extension of a torsion-free countable discrete abelian group \(H\) by a torsion-free countable discrete amenable group \(N\) and \(H\) satisfies the Kadison-Kaplansky conjecture, then so does the group \(G\) (Corollary 4.2).