Linear maps of 𝒞*-algebras preserving the absolute value

Gardner, L. Terrell

doi:10.1090/S0002-9939-1979-0537087-0

Linear maps of $\mathcal {C}^{\ast }$-algebras preserving the absolute value
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Proc. Amer. Math. Soc. 76 (1979), 271-278 Request permission

Abstract:

In order that a linear map of ${\mathcal {C}^\ast }$-algebras $\phi :\mathcal {A} \to \mathcal {B}$ 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 $\mathcal {A}$, with $xy = 0$, then $\phi (x)\phi (y) = 0$. The generalized Schwarz inequalities of Kadison and Choi are extended to the nonunital case.

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Positive linear maps of

algebras