  A Banach space $X$ is said to have the Daugavet property if every rank-one
operator $T:X\longrightarrow X$ satisfies $\|Id + T\| = 1 + \|T\|$. We give
geometric characterizations of this property in the settings of $C^*$-algebras,
$JB^*$-triples and their isometric preduals. We also show that, in these
settings, the Daugavet property passes to ultrapowers, and thus, it is
equivalent to an stronger property called the uniform Daugavet property.
