For every group $G$, the reduced group $C^* $ algebra $C^*_{red} G$ is equipped with the inner product $<a,b>=tr(ab^*)$ wherher $"tr"$ is the standard trace on group $C^*$ algebras.

For what kind of groups $G$, $C^*_{red}G$ admit a bounded skew symmetric $2-$linear map $\wedge: C^*_{red}G\times C^*_{red}G \to C^*_{red} G$ with the following properties:

1. For every $a,b \in C^*_{red}G,\; a\wedge b$ is perpendicular to both $a,b$

2)For every two independent elements $a,b\in C^*_{red} G, a\wedge b$ is a non zero element?

our question is inspired by the following paper:

https://pdfs.semanticscholar.org/1f6b/ff1e992f60eb87b35c3ceed04272fb5cc298.pdf