A note on the half-liberation operation

We propose a new approach to the half-liberation question, for the compact groups $T_N\subset G_N\subset U_N$, where $T_N={\mathbb{Z}}_2^N$. Indeed, we can construct a quantum group $T_N^{\ast }\subset G_N^{\ast }\subset U_N^{\ast }$, simply by setting $G_N^{\ast }=<G_N,T_N^{\ast }>$. We explain here how this construction fits into the known general theory of half-liberation, and we discuss as well some potential generalizations, with $T_N^{\ast }$ being replaced by more complicated objects.