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

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 : E -+ E satisfies the Daugavet equation II IdX +P II = 1 + IIP II. The analogous conclusion also holds for the alternative Daugavet property. (c) 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons .org /licenses /by -nc -nd /4 .0/).