The Bass trace conjecture for complex group rings of all discrete groups

Theorem 1.1 (24 September): for every discrete group $G$, every $x\in K_0(\mathbb CG)$ and every infinite-order $g$, $\mathrm{HS}_G(x)([g])=0$; the integral (identity-support) form for $\mathbb ZG$ is deduced in Section 9. The 5 October companion proves the $\ell^1$-Bass conjecture (BCM 2004, Conjecture 2.2): the trace of every idempotent over $\ell^1(G)$ is supported on finitely many finite-order classes, with no assembly hypothesis, which implies the complex form. Neither manuscript addresses the reduced $C^*$-algebra (the Kadison-Kaplansky question), where this release separately claims a counterexample.

Theorem 1.1 (24 September): for every discrete group $G$, every $x\in K_0(\mathbb CG)$ and every infinite-order $g$, $\mathrm{HS}_G(x)([g])=0$; the integral (identity-support) form for $\mathbb ZG$ is deduced in Section 9. The 5 October companion proves the $\ell^1$-Bass conjecture (BCM 2004, Conjecture 2.2): the trace of every idempotent over $\ell^1(G)$ is supported on finitely many finite-order classes, with no assembly hypothesis, which implies the complex form. Neither manuscript addresses the reduced $C^*$-algebra (the Kadison-Kaplansky question), where this release separately claims a counterexample.