The Daugavet equation for polynomials on C${}^{\ast }$-algebras and JB${}^{\ast }$-triples

We prove that every JB${}^{\ast }$-triple $E$ (in particular, every $C^{\ast }$-algebra) satisfying the Daugavet property also satisfies the stronger polynomial Daugavet property, that is, every weakly compact polynomial $P\text{\hspace{0.17em}⁣}:E⟶E$ satisfies the Daugavet equation $\mathrm{\parallel }{\text{id}}_E+P\mathrm{\parallel }=1+\mathrm{\parallel }P\mathrm{\parallel }$. The analogous conclusion also holds for the alternative Daugavet property.