On a positive linear map preserving absolute values
Suppose \(A\) and \(B\) are unital \(C^*\)-algebras and suppose that \(A^+= \{a^*a\); \(a\in A\}\) is a closed convex cone of positive elements in \(A\). If \(a\in A\), then \(|a|= (a^*a)^{1/2}\) is called the absolute value of \(a\). A linear map \(\varphi:A\to B\) is called positive if \(\varphi(A^+)\subset B^+\) and is called disjoint if \(xy=0\) in \(A\) implies \(\varphi(x).\varphi(y)=0\) in \(B\). \textit{L. T. Gardner} [Proc. Am. Math. Soc. 76, 271-278 (1979; Zbl 0425.46042)] proposed the following problem: Does a positive disjoint linear map of \(C^*\)-algebras preserve absolute values? In this note the authors give an affirmative answer for the case of an \(n\times n\) matrix algebra, \(n\geq 2\), and also a necessary and sufficient condition for a linear map to preserve absolute values.
scientific article; zbMATH DE number 1040499