  Analogously to the space of virtual permutations, we define projective limits
of isometries: these sequences of unitary operators are natural in the sense
that they minimize the rank norm between successive matrices of increasing
sizes. The space of virtual isometries we construct this way may be viewed as a
natural extension of the space of virtual permutations of Kerov, Olshanski and
Vershik as well as the space of virtual isometries of Neretin. We then derive
with purely probabilistic methods an almost sure convergence for these random
matrices under the Haar measure: for a coherent Haar measure on virtual
isometries, the smallest normalized eigenangles converge almost surely to a
point process whose correlation function is given by the sine kernel. This
almost sure convergence actually holds for a larger class of measures as is
proved by Borodin and Olshanski. We give a different proof, probabilist in the
sense that it makes use of martingale arguments and shows how the eigenangles
interlace when going from dimension n to n+1. Our method also proves that for
some universal constant epsilon>0, the rate of convergence is almost surely
dominated by n^{-epsilon} when the dimension n goes to infinity.
