Hiai, Fumio; Petz, Dénes Properties of free entropy related to polar decomposition. (English) Zbl 0934.46063 Commun. Math. Phys. 202, No. 2, 421-444 (1999). 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. Cited in 3 Documents MSC: 46L54 Free probability and free operator algebras 46L55 Noncommutative dynamical systems Keywords:free entropies; non-selfadjoint random variables; unitary random variables; Voiculescu’s free analysis; subadditivity PDFBibTeX XMLCite \textit{F. Hiai} and \textit{D. Petz}, Commun. Math. Phys. 202, No. 2, 421--444 (1999; Zbl 0934.46063) Full Text: DOI