Spectral relations and unitary mixing in semifinite von Neumann algebras (Q1100695)

Let (\({\mathcal M},\tau)\) be a pair of a semi-finite von Neumann algebra and a faithful normal semi-finite trace. The topological algebra \(\tilde {\mathcal M}\) of \(\tau\)-measurable operators affiliated with \({\mathcal M}\) is introduced in the sense of \textit{E. Nelson} [J. Funct. Anal. 15, 103-116 (1974; Zbl 0292.46030)] and \textit{Terp} [Copenhagen Univ. Notes (1981)] together with several spectral relations on \(\tilde {\mathcal M}\) in terms of generalized s-numbers [see \textit{T. Fack} and \textit{H. Kosaki}, Pac. J. Math. 123, 269-300 (1986; Zbl 0617.46063)]. Then the condition of unitary mixing is characterized in terms of spectral relations such as majorization or submajorization between positive elements of \(\tilde {\mathcal M}\). The main result is that, for positive x and y in \(L^ 1({\mathcal M},\tau)\), x is in the \(\| \|_ 1\)-closed convex hull of the unitary orbit of y if and only if ex is majorized by ey for every central projection e.