In this paper, we deal with the problem of characterizing linear maps compressing the numerical range. A counterexample is given to show that such a map need not be a Jordan $*$-homomorphism in general even if the $C^*$-algebras are commutative. Next, under an auxiliary condition we show that such a map is a Jordan $*$-homomorphism.
Aschwag Fahad Albideewi. Mohamed Mabruk. "On maps compressing the numerical range between $C^*$-algebras." Adv. Oper. Theory 2 (2) 108 - 113, Spring 2017. https://doi.org/10.22034/aot.1612-1067