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

Title & Authors
MAPPING PRESERVING NUMERICAL RANGE OF OPERATOR PRODUCTS ON C*-ALGEBRAS
MABROUK, MOHAMED;

Abstract
Let $\small{\mathcal{A}}$ and $\small{\mathcal{B}}$ be two unital $\small{C^*}$-algebras. Denote by W(a) the numerical range of an element $\small{a{\in}\mathcal{A}}$. We show that the condition W(ax) = W(bx), $\small{{\forall}x{\in}\mathcal{A}}$ implies that a = b. Using this, among other results, it is proved that if $\small{{\phi}}$ : $\small{\mathcal{A}{\rightarrow}\mathcal{B}}$ is a surjective map such that $\small{W({\phi}(a){\phi}(b){\phi}(c))=W(abc)}$ for all a, b and $\small{c{\in}\mathcal{A}}$, then $\small{{\phi}(1){\in}Z(B)}$ and the map $\small{{\psi}={\phi}(1)^2{\phi}}$ is multiplicative.
Keywords
$\small{C^*}$-algebras;numerical range;preserving the numerical range;
Language
English
Cited by
1.
Numerical radius characterizations of elements in -algebras, Linear and Multilinear Algebra, 2017, 1
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