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Some inequalities for positive linear maps. (English) Zbl 1252.15024

Let M n () be the algebra of all n×n complex matrices and let φ:M n ()M n () be a positive unital map. The authors prove that if AM n (), then

φ(A * A)-φ(A) * φ(A)inf z A-z·

Moreover, the authors give many surprisingly and strong consequences of this inequality. Also, they demonstrate the connection between this inequality and several old and new results.

MSC:
15A45Miscellaneous inequalities involving matrices
15A60Applications of functional analysis to matrix theory
46L53Noncommutative probability and statistics
47A63Operator inequalities
47C15Operators in C * - or von Neumann algebras
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