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

For every group $G$, the reduced group $C^*$-algebra $C^*_{red}G$ is equipped with the inner product $\langle a,b\rangle=tr(ab^*)$ where "$tr$" 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 nonzero element?

Our question is inspired by the following paper:

W. S. Massey, Cross Products of Vectors in Higher Dimensional Euclidean Spaces. The American Mathematical Monthly, Vol. 90, No. 10 (Dec., 1983), pp. 697-701.