The alternative Daugavet property of C *‐algebras and JB *‐triples

Abstract

A Banach space X is said to have the alternative Daugavet property if for every (bounded and linear) rank-one operator T : X → X there exists a modulus one scalar ω such that Id + ωT = 1 + T . We give geometric characterizations of this property in the setting of C * -algebras, JB * -triples, and of their isometric preduals.For a real or complex Banach space X, we write X * for its topological dual and L(X) for the Banach algebra of bounded linear operators on X. We denote by T the set of modulus-one scalars.A Banach space X is said to have the alternative Daugavet property if the norm equalityholds for every rank-one operator T ∈ L(X) [38]. In such a case, all weakly compact operators on X also satisfy (aDE) (see [38, Theorem 2.2]). It is clear that a Banach space X has the alternative Daugavet property whenever X * has, but the reverse result does not hold [38, Example 4.4].Observe that (aDE) for an operator T just means that there exists a modulus-one scalar ω such that the operator S = ωT satisfies the well-known Daugavet equationTherefore, the Daugavet property (i.e. every rank-one, equivalently, every weakly compact, operator satisfies (DE) [25, Theorem 2.3]) implies the alternative Daugavet property. Examples of spaces having the Daugavet property are C(K) and L 1 (µ), provided that K is perfect and µ does not have any atoms (see [45] for an elementary approach), and certain function algebras such as the disk algebra A(D) or the algebra of bounded analytic functions H ∞ [46,48]. A good introduction to the Daugavet equation is given in the books [3,4] and the state-of-the-art on the subject can be found in the papers [25,47]. For very recent results we refer the reader to [6,8,26,27,41] and references therein.

The Daugavet equation for polynomials

“…The above norm equality is now known as the alternative Daugavet equation [22], and it is satisfied by all bounded linear operators on C(K) and L 1 (µ), K and µ arbitrary. We refer the reader to [16,21,22] and references therein for background. The aim of this paper is to study the Daugavet equation and the alternative Daugavet equation for polynomials in Banach spaces.…”

Positive and negative results on the numerical index of Banach spaces and duality

“…Good references for the Daugavet property are the books [1,2] and the papers [16,24]. Information on the alternative Daugavet property can be found in [19,20].…”

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

“…This property was formally introduced and deeply studied in [46] (some related ideas had appeared before). We also refer the reader to [4,35,44] for instance, for more information and background and for the relationship between the ADP and the study of the numerical index of Banach spaces.…”

“…As this manuscript will deal with C * -algebras and JB * -triples, it makes sense to present the characterizations of the DPr and the ADP for these spaces given in [11,44,46,48]. We refer to Section 3 for the definition of the involved concepts.…”