The Alternative Daugavet Property of $C^{\ast }$-algebras and $J{B}^{\ast }$-triples

Abstract

A Banach space $X$ is said to have the alternative Daugavet property if for every (bounded and linear) rank-one operator $T:X⟶X$ there exists a modulus one scalar $\omega$ such that $\Vert Id+\omega T\Vert =1+\Vert T\Vert$. We give geometric characterizations of this property in the setting of $C^{\ast }$-algebras, $J{B}^{\ast }$-triples and their isometric preduals.