Jensen’s inequality in semi-finite von Neumann algebras. (English) Zbl 0718.46026

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 \(xy,yx\in L^ 1({\mathcal M},\tau)\).


46L10 General theory of von Neumann algebras
46L51 Noncommutative measure and integration
46L53 Noncommutative probability and statistics
46L54 Free probability and free operator algebras