Brown, Lawrence G.; Kosaki, Hideki Jensen’s inequality in semi-finite von Neumann algebras. (English) Zbl 0718.46026 J. Oper. Theory 23, No. 1, 3-19 (1990). Operator concavity of a function f: \({\mathbb{R}}_+\to {\mathbb{R}}\), with \(f(0)=0\), which ensures that Hansen’s inequality \(a^*f(x)a\leq f(a^*xa)\) holds for any positive operator x and any contraction a, is a very strong condition on f. Here it is shown that for the inequality inside a trace \(\tau\) (on a semi-finite von Neumann algebra \({\mathcal M})\) to hold it suffices that f is a continuous concave function on \({\mathbb{R}}\), with \(f(0)=0\). A consequence of this result is a simple proof of \(\tau (xy)=\tau (yx)\) for x,y with \(xy,yx\in L^ 1({\mathcal M},\tau)\). Reviewer: H.Schröder (Augsburg) Cited in 1 ReviewCited in 45 Documents MSC: 46L10 General theory of von Neumann algebras 46L51 Noncommutative measure and integration 46L53 Noncommutative probability and statistics 46L54 Free probability and free operator algebras Keywords:Operator concavity; Hansen’s inequality; positive operator; contraction; trace; semi-finite von Neumann algebra PDFBibTeX XMLCite \textit{L. G. Brown} and \textit{H. Kosaki}, J. Oper. Theory 23, No. 1, 3--19 (1990; Zbl 0718.46026)