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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.

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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