On maps compressing the numerical range between $C^*$-algebras

Abstract

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.