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Jensen’s inequality for positive contractions on operator algebras. (English) Zbl 0622.46044
Let $$\tau$$ be a normal semifinite trace on a von Neumann algebra, and let f be a continuous convex function on the interval [0,$$\infty)$$ with $$f(0)=0$$. For a positive element a of the algebra and a positive contraction $$\alpha$$ on the algebra, the following inequality is obtained: $$\tau$$ (f($$\alpha$$ (a)))$$\leq \tau (\alpha (f(a)))$$. If $$\alpha$$ is unital then the condition $$f(0)=0$$ is not necessary.

##### MSC:
 46L51 Noncommutative measure and integration 46L53 Noncommutative probability and statistics 46L54 Free probability and free operator algebras 26A51 Convexity of real functions in one variable, generalizations 47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.)
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