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:
- 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.
