Mapping preserving numerical range of operator products on ${C}^*$-algebras
Bull. Korean Math. Soc. 2015 Vol. 52, No. 6, 1963-1971
https://doi.org/10.4134/BKMS.2015.52.6.1963
Published online November 30, 2015
Mohamed Mabrouk
Faculty of Sciences
Abstract : Let $\ala$ and $\alb$ be two unital $C^*$-algebras. Denote by $W(a)$ the numerical range of an element $a\in\ala$. We show that the condition $W(ax)=W(bx), \forall x\in\ala$ implies that $a=b$. Using this, among other results, it is proved that if $\phi: \ala\rightarrow\alb$ is a surjective map such that $W(\phi(a)\phi(b)\phi(c)) = W(abc)$ for all $a, b$ and $c \in\ala$, 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
MSC numbers : 15A86, 46L05, 47A12, 47B49
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