Abstract
In order that a linear map of C ∗ {\mathcal {C}^\ast } -algebras ϕ : A → B \phi :\mathcal {A} \to \mathcal {B} preserve absolute values, it is necessary and sufficient that it be 2-positive and preserve zero products of positive elements: if x and y are positive in A \mathcal {A} , with x y = 0 xy = 0 , then ϕ ( x ) ϕ ( y ) = 0 \phi (x)\phi (y) = 0 . The generalized Schwarz inequalities of Kadison and Choi are extended to the nonunital case.
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