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Properties of free entropy related to polar decomposition. (English) Zbl 0934.46063

Summary: The free entropies \(\widehat\chi(a_1,\dots, a_N)\) of non-selfadjoint random variables and \(\chi_u(u_1,\dots, u_N)\) of unitary random variables are introduced and discussed by the methods of Voiculescu’s free analysis. The additivity \(\chi_u(u_1,\dots, u_N)= \sum_i\chi_u(u_i)\) is shown to be equivalent to freeness. The relation among \(\widehat\chi\), \(\chi_u\) and \(\chi\) is investigated in the case when \(a_i= u_ih_i\) is the polar decomposition. The subadditivity \(\widehat\chi(a_1,\dots, a_N)\leq \chi_u(u_1,\dots, u_N)+ \chi(h^2_1,\dots, h^2_N)+\text{constant}\) is proven and applications to some maximization problems for \(\widehat\chi\) are given.

MSC:

46L54 Free probability and free operator algebras
46L55 Noncommutative dynamical systems
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