Klyachko's theorem in semi-finite von Neumann algebras
Given positive \(\tau\)-measurable operators \(A,B\) over a semifinite von Neumann algebra \(M\), the author finds a sufficient condition over measurable sets \(I,J,K\) so that \[ \int_K\mu_s(A+B)\,ds\leq\int_I\mu_s(A)\,ds+\int_J\mu_s(B)\,ds. \] Here, \(\mu_s(A)\) is the spectral scale of \(A\) [\textit{D.\,Petz}, J.~Math.\ Anal.\ Appl.\ 109, 74--82 (1985; Zbl 0655.47032)]. This is inspired by the well-known and celebrated Klyachko--Knutson-Tao results [\textit{A.\,Klyachko}, Sel.\ Math., New Ser.\ 4, No.\,3, 419--445 (1998; Zbl 0915.14010); \textit{A.\,Knutson} and \textit{T.\,Tao}, J.~Am.\ Math.\ Soc.\ 12, No.\,4, 1055--1090 (1999; Zbl 0944.05097)], which imply a complete characterization of the situation above in the finite-dimensional case. Other complete characterizations appear in the work of \textit{S.\,Friedland} (for trace-class operators, in [Linear Algebra Appl.\ 319, No.\,~1--3, 3--22 (2000; Zbl 0979.47016)]) and \textit{H.\,Bercovici} and \textit{W.\,S.\thinspace Li} (for II\(_1\)-factors, in [Proc.\ Am.\ Math.\ Soc.\ 134, No.\,1, 75--80 (2006; Zbl 1086.15019)]).
scientific article; zbMATH DE number 5343281