  The partial isometries of $\mathbb R^N,\mathbb C^N$ form compact semigroups
$\widetilde{O}_N,\widetilde{U}_N$. We discuss here the liberation question for
these semigroups, and for their discrete versions
$\widetilde{H}_N,\widetilde{K}_N$. Our main results concern the construction of
half-liberations
$\widetilde{H}_N^\times,\widetilde{K}_N^\times,\widetilde{O}_N^\times,\widetilde{U}_N^\times$
and of liberations
$\widetilde{H}_N^+,\widetilde{K}_N^+,\widetilde{O}_N^+,\widetilde{U}_N^+$. We
include a detailed algebraic and probabilistic study of all these objects,
justifying our "half-liberation" and "liberation" claims.
