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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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