The Radon-Nikodym theorem for non-commutative \(L^p\)-spaces (Q1025764)

In his doctoral dissertation [``Non-commutative \(L^p\)-spaces constructed by the complex interpolation method'' (Tohoku Mathematical Publications 9; Sendai:\ Tohoku Univ.) (1998; Zbl 0921.46070)], followed by a series of papers on the same subject, the author used complex interpolation to define families of noncommutative \(L^p\)-spaces over an arbitrary von Neumann algebra with a weight. These constructions generalized those of Kosaki, who defined them for a \(\sigma\)-finite von Neumann algebra with a state, and that of Terp, who dealt with a general algebra, but considered only the symmetric version. Since all these spaces are isometrically isomorphic to Haagerup spaces, they do not depend on the choice of the weight. Nevertheless, obtaining an explicit formula for the isometry between \(L^p_{(\alpha)}({\mathcal M},\varphi)\) and \(L^p_{(\alpha)}({\mathcal M},\psi)\) is nontrivial and is the subject of the present paper. Complicated calculations and a clever use of the balanced weight \(\phi \oplus \psi\) on the von Neumann algebra \(M_2({\mathcal M})\) lead to an isometry \(U^{\psi,\varphi}_{p,(\alpha)}\) that is natural in the sense of being the inverse of \(U^{\varphi,\psi}_{p,(\alpha)}\) and satisfying the chain rule \(U^{\theta,\psi}_{p,(\alpha)}U^{\psi,\varphi}_{p,(\alpha)}=U^{\theta,\varphi}_{p,(\alpha)}\).