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:E⟶E satisfies the Daugavet equation ‖IdX+P‖=1+‖P‖. The analogous conclusion also holds for the alternative Daugavet property.
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