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http://dx.doi.org/10.4134/BKMS.2015.52.6.1963

MAPPING PRESERVING NUMERICAL RANGE OF OPERATOR PRODUCTS ON C*-ALGEBRAS  

MABROUK, MOHAMED (Department of Mathematics College of Applied Sciences and Department of Mathematics Faculty of Sciences)
Publication Information
Bulletin of the Korean Mathematical Society / v.52, no.6, 2015 , pp. 1963-1971 More about this Journal
Abstract
Let $\mathcal{A}$ and $\mathcal{B}$ be two unital $C^*$-algebras. Denote by W(a) the numerical range of an element $a{\in}\mathcal{A}$. We show that the condition W(ax) = W(bx), ${\forall}x{\in}\mathcal{A}$ implies that a = b. Using this, among other results, it is proved that if ${\phi}$ : $\mathcal{A}{\rightarrow}\mathcal{B}$ is a surjective map such that $W({\phi}(a){\phi}(b){\phi}(c))=W(abc)$ for all a, b and $c{\in}\mathcal{A}$, then ${\phi}(1){\in}Z(B)$ and the map ${\psi}={\phi}(1)^2{\phi}$ is multiplicative.
Keywords
$C^*$-algebras; numerical range; preserving the numerical range;
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