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

David Cabezas, Miguel Martín, Antonio M. Peralta

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.

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

Abstract

We prove that every JB-triple (in particular, every -algebra) satisfying the Daugavet property also satisfies the stronger polynomial Daugavet property, that is, every weakly compact polynomial satisfies the Daugavet equation . The analogous conclusion also holds for the alternative Daugavet property.
Paper Structure (4 sections, 13 theorems, 26 equations)

This paper contains 4 sections, 13 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Revisiting the Daugavet equation for polynomials
  3. A closer look at the theory of C$^*$-algebras and JB$^*$-triples
  4. Main results

Key Result

Theorem 1.1

$$ (a). Let $X$ be a C$^*$-algebra. (b). Let $X$ be a JB$^*$-triple.

Theorems & Definitions (19)

  • Theorem 1.1: BeMar2005Mar2008MarOikh2004Oik2002
  • Theorem 1.2: Main result
  • Proposition 2.1: cgmm2
  • Proposition 2.2: cgmm2
  • Theorem 2.3
  • Lemma 2.4: ShvydkoyJFA2000
  • proof : Proof of Theorem \ref{['prop:generalDPr-weaklycontinouspolynomials']}
  • Lemma 3.1
  • Remark 3.2
  • Lemma 3.3
  • ...and 9 more