The Daugavet equation for polynomials on C$^*$-algebras and JB$^*$-triples

Abstract: We prove that every JB$^*$-triple $E$ (in particular, every $C^*$-algebra) satisfying the Daugavet property also satisfies the stronger polynomial Daugavet property, that is, every weakly compact polynomial $P\colon E \longrightarrow E$ satisfies the Daugavet equation $|\hbox{id}_{E} + P| = 1+|P|$. The analogous conclusion also holds for the alternative Daugavet property.