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

TL;DR

This paper proves that JB$^*$-triples and $C^*$-algebras with the Daugavet property also satisfy a stronger polynomial version, extending the class of spaces known to have the Daugavet equation for polynomials.

Contribution

It establishes that spaces with the Daugavet property also satisfy the polynomial Daugavet property, including for weakly compact polynomials, in the context of JB$^*$-triples and $C^*$-algebras.

Findings

JB$^*$-triples with Daugavet property satisfy polynomial Daugavet property

The result extends to spaces with the alternative Daugavet property

Weakly compact polynomials satisfy the Daugavet equation in these spaces

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.