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 $T_N^*\subset G_N^*\subset U_N^*$, simply by setting $G_N^*=<G_N,T_N^*>$. We explain here how this construction fits into the known general theory of half-liberation, and we discuss as well some potential generalizations, with $T_N^*$ being replaced by more complicated objects.