[The Kaplansky conjecture](https://en.wikipedia.org/wiki/Kaplansky%27s_conjectures)  and   Kaplanski-Kadison conjecture are classical conjectures about non existence of non-trivial idempotents in group algebra $\mathbb{C}\Gamma $ or the reduced group algebra $C_{\mathrm{red}}^*\Gamma$  where $\Gamma$ is  a torsion free group. 

 On the other hand, the value of  **trace** of  an idempotent plays a  crucial role in investigation of idempotent problem in certain $C^*$-algebra or group algebra. The rationality or integrality  of $\tau(e)$  or the range of  change of $\tau(e)$... etc. See [Idempotents in complex group rings:
theorems of Zalesskii and Bass revisited
][1] 


Now we notice that trace is a **cocycle** of the Hochschild complex. 

>So a  natural  question is that what type of other **higher dimensional** cocycles have been examined in the context of  idempotent problems for  group algebra or  reduce group algebras.


As a related point we know from page 21 of [Non-Commutative Geometry](https://alainconnes.org/wp-content/uploads/book94bigpdf.pdf) that $\tau(e,e,e)$ is constant on every curve of  idempotents where $\tau$ is  a cyclic 2-cocycle.

>But is there any precise case of torsion-free group $\Gamma$ for which the standard trace does not work but a higher order cocycle  works to prove that the  corresponding algebra does not have any non trivial idempotent?


  [1]: https://www.heldermann-verlag.de/jlt/jlt08/VALLAT.PDF