Properties of free entropy related to polar decomposition

The free entropies <(chi)over cap>(a(1), ..., a(N)) of non-selfadjoint rand om variables and chi(u)(u(1), ..., u(N)) of unitary random variables are in troduced and discussed by the methods of Voiculescu's free analysis. The ad ditivity chi(u)(u(1), ..., u(N)) = Sigma(i) chi(u)(u(i)) is shown to be equ ivalent to freeness. The relation among <(chi)over cap>, chi(u) and chi is investigated in the case when a(i) = u(i)h(i) is the polar decomposition. T he subadditivity <(chi)over cap>(a(1), ..., a(N)) less than or equal to chi (u)(u(1), ..., u(N)) + chi(h(1)(2), ..., h(N)(2)) + constant is proven and applications to some maximization problems for <(chi)over cap> are given.