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