The partial isometries of R-N, C-N form compact semigroups (O) over tilde (N), (U) over tilde (N). We discuss here the liberation question for these semigroups, and for their discrete versions (H) over tilde (N), (K) over tilde (N). Our main results concern the construction of half- liberations (H) over tilde (x)(N), (K) over tilde (x)(N), (O) over tilde (x)(N), (U) over tilde (x)(N) and ofliberations (H) over tilde (+)(N), (K) over tilde (+)(N), (O) over tilde (+)(N), (U) over tilde (+)(N). We include a detailed algebraic and probabilistic study of all these objects, justifying our "half-liberation" and "liberation" claims.
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