We study the microstate free entropy χproj(_p_1 , . . . , pn) of projections, and establish its basic properties similar to the self-adjoint variable case. Our main contribution is to characterize the pair-block freeness of projections by the additivity of χproj (Theorem 4.1), in the proof of which a transportation cost inequality plays an important role. We also brieﬂy discuss the free pressure in relation to χproj.
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Fumio Hiai, Yoshimichi Ueda, Notes on Microstate Free Entropy of Projections. Publ. Res. Inst. Math. Sci. 44 (2008), no. 1, pp. 49–89DOI 10.2977/PRIMS/1207921076