The algebraic structure of quantum partial isometries

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