Polynomial Daugavet Center Overview

• The polynomial Daugavet center is a nonzero homogeneous polynomial Q that satisfies the norm-additive identity with every rank-one polynomial, thereby extending the classical Daugavet property.
• It leverages the weak polynomial topology and the duality with symmetric projective tensor products to bridge homogeneous polynomials and tensor geometry.
• The framework unifies earlier results by proving the equivalence of linear and polynomial Daugavet properties, influencing studies in L1-preduals, JB*-triples, and C*-algebras.

A polynomial Daugavet center is a nonzero polynomial $Q\in \mathcal P(X,Y)$ such that

$\|Q+P\|=\|Q\|+\|P\|$

for every rank-one polynomial $P\in \mathcal P(X,Y)$; in this case the same norm equality holds for all weakly compact polynomials as well (Dantas et al., 8 Jul 2025). The notion extends the linear Daugavet center, where $Q$ is replaced by a nonzero bounded linear operator $G:X\to Y$ satisfying the same equation for rank-one operators (Bosenko, 2010). When $Q=\mathrm{Id}_X$, one obtains the polynomial Daugavet property, so the subject lies at the intersection of Daugavet theory, weakly compact polynomials, and the geometry of symmetric projective tensor products, where

$$\big(\widehat\otimes_{\pi,s}^N X\big)^* \cong \mathcal P(^N X)$$

provides the standard bridge between homogeneous polynomials and tensor geometry (Zoca et al., 2019).

1. Definition and basic framework

The linear antecedent is the Daugavet center. A linear continuous nonzero operator $G:X\to Y$ is a Daugavet center if every rank-one operator $T:X\to Y$ satisfies

$\|G+T\|=\|G\|+\|T\|.$

A Banach space $\|Q+P\|=\|Q\|+\|P\|$0 has the classical Daugavet property exactly when $\|Q+P\|=\|Q\|+\|P\|$1 is a Daugavet center (Bosenko, 2010).

In the polynomial setting, a rank-one polynomial has the form $\|Q+P\|=\|Q\|+\|P\|$2, where $\|Q+P\|=\|Q\|+\|P\|$3 is scalar-valued and $\|Q+P\|=\|Q\|+\|P\|$4. The polynomial Daugavet property is the requirement that every weakly compact polynomial $\|Q+P\|=\|Q\|+\|P\|$5 satisfy

$\|Q+P\|=\|Q\|+\|P\|$6

and $\|Q+P\|=\|Q\|+\|P\|$7 has the polynomial Daugavet property if and only if $\|Q+P\|=\|Q\|+\|P\|$8 is a polynomial Daugavet center (Martin et al., 2020, Dantas et al., 8 Jul 2025).

A geometric characterization replaces linear slices by polynomial data. For a Banach space $\|Q+P\|=\|Q\|+\|P\|$9, the polynomial Daugavet property is equivalent to the following: for every $P\in \mathcal P(X,Y)$0, every $P\in \mathcal P(X,Y)$1, and every norm-one scalar polynomial $P\in \mathcal P(X,Y)$2, there exist $P\in \mathcal P(X,Y)$3 and $P\in \mathcal P(X,Y)$4 such that

$P\in \mathcal P(X,Y)$5

The analogous characterization for a polynomial Daugavet center $P\in \mathcal P(X,Y)$6 replaces $P\in \mathcal P(X,Y)$7 by an arbitrary $P\in \mathcal P(X,Y)$8 and $P\in \mathcal P(X,Y)$9 by $Q$0 (Dantas et al., 8 Jul 2025).

2. Geometric characterizations and the weak polynomial topology

A decisive development was the replacement of weak topology by weak polynomial topology. The weak polynomial topology on $Q$1 is the smallest topology making all scalar polynomials in $Q$2 continuous; equivalently, a net $Q$3 converges to $Q$4 if

$Q$5

Its bidual analogue is the polynomial-star topology defined through Aron–Berner extensions (Dantas et al., 8 Jul 2025).

The key technical theorem establishes a polynomial version of Shvidkoy’s lemma: if $Q$6 has the Daugavet property, then for every $Q$7 and $Q$8 there exists a net $Q$9 such that $G:X\to Y$0 in the weak polynomial topology and

$G:X\to Y$1

Equivalently, for every $G:X\to Y$2 and $G:X\to Y$3, the set

$G:X\to Y$4

is dense in $G:X\to Y$5 for the relative weak polynomial topology (Dantas et al., 8 Jul 2025).

This theorem yields the general equivalence

$G:X\to Y$6

thereby solving a longstanding open problem (Dantas et al., 8 Jul 2025). It also implies that every linear Daugavet center is automatically a polynomial Daugavet center. In the same vein, the weak operator Daugavet property implies its polynomial counterpart, again by upgrading weak density statements to weak polynomial density statements (Dantas et al., 8 Jul 2025).

Historically, this result completed a pattern already visible in several special classes. Earlier work had proved the implication from the Daugavet property to the polynomial Daugavet property for $G:X\to Y$7-preduals, spaces of Lipschitz functions, JB$G:X\to Y$8-triples, and $G:X\to Y$9-algebras (Martin et al., 2020, Cabezas et al., 2022). The 2025 theorem shows that no class restriction is needed (Dantas et al., 8 Jul 2025).

3. Symmetric tensor products and the polynomial viewpoint

The tensorial framework is central because continuous $Q=\mathrm{Id}_X$0-homogeneous polynomials are dual to symmetric projective tensor powers. For a Banach space $Q=\mathrm{Id}_X$1,

$Q=\mathrm{Id}_X$2

is the completion of the algebraic symmetric tensor product under the norm

$Q=\mathrm{Id}_X$3

and

$Q=\mathrm{Id}_X$4

isometrically (Zoca et al., 2019).

This duality turns questions about polynomials into questions about Daugavet geometry of symmetric tensors. An early result showed that if $Q=\mathrm{Id}_X$5 has the operator Daugavet property, then $Q=\mathrm{Id}_X$6 has an octahedral norm for every $Q=\mathrm{Id}_X$7, and that $Q=\mathrm{Id}_X$8 has the Daugavet property whenever $Q=\mathrm{Id}_X$9 is a compact Hausdorff space without isolated points (Zoca et al., 2019). These were described as the first nontrivial examples of symmetric projective tensor products with the Daugavet property (Zoca et al., 2019).

The subsequent refinement is stronger. If $$\big(\widehat\otimes_{\pi,s}^N X\big)^* \cong \mathcal P(^N X)$$0 has the polynomial weak operator Daugavet property, then every symmetric projective tensor power $$\big(\widehat\otimes_{\pi,s}^N X\big)^* \cong \mathcal P(^N X)$$1 has the weak operator Daugavet property and hence the Daugavet property (Martin et al., 2020). This applies in particular to $$\big(\widehat\otimes_{\pi,s}^N X\big)^* \cong \mathcal P(^N X)$$2-preduals with the Daugavet property and to vector-valued spaces $$\big(\widehat\otimes_{\pi,s}^N X\big)^* \cong \mathcal P(^N X)$$3 with $$\big(\widehat\otimes_{\pi,s}^N X\big)^* \cong \mathcal P(^N X)$$4 atomless (Martin et al., 2020). Consequently, symmetric tensor powers provide a canonical supply of Daugavet spaces whose duals are polynomial spaces.

From the polynomial-center perspective, the significance is structural rather than terminological. Since $$\big(\widehat\otimes_{\pi,s}^N X\big)^* \cong \mathcal P(^N X)$$5 is the dual of a Daugavet space in these cases, the unit ball of the polynomial space inherits the corresponding dual diameter-two geometry, and the ambient tensor product behaves as a Daugavet center for the homogeneous polynomial theory attached to $$\big(\widehat\otimes_{\pi,s}^N X\big)^* \cong \mathcal P(^N X)$$6 (Zoca et al., 2019, Martin et al., 2020).

4. Principal classes of polynomial Daugavet centers

Several major classes are now known to satisfy the polynomial Daugavet property, hence to be polynomial Daugavet centers via the identity. The most important classes are summarized below.

Class	Hypothesis	Conclusion
$$\big(\widehat\otimes_{\pi,s}^N X\big)^* \cong \mathcal P(^N X)$$7-preduals	Daugavet property	Polynomial Daugavet property; all $$\big(\widehat\otimes_{\pi,s}^N X\big)^* \cong \mathcal P(^N X)$$8 have the Daugavet property (Martin et al., 2020)
$$\big(\widehat\otimes_{\pi,s}^N X\big)^* \cong \mathcal P(^N X)$$9	Daugavet property	Polynomial Daugavet property (Martin et al., 2020)
JB$G:X\to Y$0-triples	Daugavet property	Polynomial Daugavet property (Cabezas et al., 2022)
$G:X\to Y$1-algebras	Diffuse	Polynomial Daugavet property (Cabezas et al., 2022)
$G:X\to Y$2	Base algebra $G:X\to Y$3 or range $G:X\to Y$4 has polynomial Daugavet property	$G:X\to Y$5 has polynomial Daugavet property (Lee et al., 2023)

For $G:X\to Y$6-preduals, the theorem is particularly strong: the Daugavet property and the polynomial Daugavet property are equivalent, and the same is true for spaces of Lipschitz functions (Martin et al., 2020). The proofs pass through localized Daugavet geometry, weak operator variants, and tensor stability.

For JB$G:X\to Y$7-triples, every weakly compact polynomial $G:X\to Y$8 satisfies

$G:X\to Y$9

whenever $T:X\to Y$0 has the Daugavet property; the analogous conclusion also holds for the alternative Daugavet property and the alternative polynomial Daugavet property (Cabezas et al., 2022). Since a $T:X\to Y$1-algebra has the Daugavet property exactly when it is diffuse, diffuse $T:X\to Y$2-algebras are polynomial Daugavet centers in the identity sense (Cabezas et al., 2022).

Function spaces $T:X\to Y$3 exhibit a precise max formula. Their polynomial Daugavetian index satisfies

$T:X\to Y$4

so $T:X\to Y$5 has the polynomial Daugavet property if and only if either the base algebra $T:X\to Y$6 or the range space $T:X\to Y$7 has the polynomial Daugavet property (Lee et al., 2023). As a consequence, for infinite-dimensional uniform algebras the polynomial Daugavet property, the Daugavet property, the diametral diameter two properties, and property $T:X\to Y$8 are equivalent (Lee et al., 2023).

5. Localized geometry and quantitative invariants

The modern theory also has a localized side. Daugavet-points, $T:X\to Y$9-points, and relative Daugavet-points encode pointwise versions of the global Daugavet geometry. In $\|G+T\|=\|G\|+\|T\|.$0-preduals, Daugavet-points and $\|G+T\|=\|G\|+\|T\|.$1-points admit characterizations in terms of extreme points and weak$\|G+T\|=\|G\|+\|T\|.$2 accumulation points of $\|G+T\|=\|G\|+\|T\|.$3, and these characterizations feed directly into the polynomial theory for $\|G+T\|=\|G\|+\|T\|.$4-preduals (Martin et al., 2020). Relative Daugavet-points localize the behavior inside a supporting slice and sit strictly between Daugavet-points and $\|G+T\|=\|G\|+\|T\|.$5-points (Abrahamsen et al., 2023).

A quantitative refinement is given by the Daugavet constant $\|G+T\|=\|G\|+\|T\|.$6 and the $\|G+T\|=\|G\|+\|T\|.$7-constant $\|G+T\|=\|G\|+\|T\|.$8 of a point $\|G+T\|=\|G\|+\|T\|.$9. For $\|Q+P\|=\|Q\|+\|P\|$00,

$\|Q+P\|=\|Q\|+\|P\|$01

These constants satisfy

$\|Q+P\|=\|Q\|+\|P\|$02

and they are linked to slice geometry by

$\|Q+P\|=\|Q\|+\|P\|$03

(Choi et al., 2023). They are localized counterparts of global Daugavet indices of thickness.

The global indices $\|Q+P\|=\|Q\|+\|P\|$04, $\|Q+P\|=\|Q\|+\|P\|$05, and $\|Q+P\|=\|Q\|+\|P\|$06 quantify how close a space is to the Daugavet property, with

$\|Q+P\|=\|Q\|+\|P\|$07

and

$\|Q+P\|=\|Q\|+\|P\|$08

(Haller et al., 2020). For norm-one weakly compact operators $\|Q+P\|=\|Q\|+\|P\|$09, one has the lower bound

$\|Q+P\|=\|Q\|+\|P\|$10

which makes these indices operator-theoretic rather than merely geometric (Haller et al., 2020).

These localized and quantitative frameworks are not themselves definitions of polynomial Daugavet centers, but they control the slice structure, diameter-two behavior, and dentability phenomena from which polynomial Daugavet theorems are derived. In particular, the transition from weak topology to weak polynomial topology in the 2025 equivalence theorem is best understood as a polynomial upgrade of this slice-based local geometry (Dantas et al., 8 Jul 2025).

6. Variants, historical development, and open directions

Before the full equivalence theorem, polynomial Daugavet theory developed through class-specific results. The implication from the Daugavet property to the polynomial Daugavet property was known for $\|Q+P\|=\|Q\|+\|P\|$11-preduals and spaces of Lipschitz functions (Martin et al., 2020), and for JB$\|Q+P\|=\|Q\|+\|P\|$12-triples and $\|Q+P\|=\|Q\|+\|P\|$13-algebras (Cabezas et al., 2022). Function spaces $\|Q+P\|=\|Q\|+\|P\|$14 were understood through the polynomial Daugavetian index and the max formula for $\|Q+P\|=\|Q\|+\|P\|$15 (Lee et al., 2023). The 2025 result unified these strands by proving the equivalence in complete generality and by showing that every linear Daugavet center is a polynomial Daugavet center (Dantas et al., 8 Jul 2025).

Two adjacent nonlinear theories should be distinguished from the polynomial one. The $\|Q+P\|=\|Q\|+\|P\|$16-Daugavet property for constant $\|Q+P\|=\|Q\|+\|P\|$17 $\|Q+P\|=\|Q\|+\|P\|$18-convex generalized function spaces is governed by the equation

$\|Q+P\|=\|Q\|+\|P\|$19

which is a nonlinear $\|Q+P\|=\|Q\|+\|P\|$20-type analogue rather than a theory of polynomials (Perez et al., 2011). The almost Daugavet property is another relaxation, defined through a norming subspace of the dual and equivalent, in the separable case, to thickness $\|Q+P\|=\|Q\|+\|P\|$21 and to the existence of a canonical $\|Q+P\|=\|Q\|+\|P\|$22-type sequence (Lücking, 2010). Both frameworks are part of the broader Daugavet landscape, but neither is a substitute for the polynomial Daugavet center.

One open direction remains explicit in the general theory. It is unknown whether a polynomial $\|Q+P\|=\|Q\|+\|P\|$23 that satisfies

$\|Q+P\|=\|Q\|+\|P\|$24

for all rank-one linear operators $\|Q+P\|=\|Q\|+\|P\|$25 must already be a polynomial Daugavet center in the full sense, that is, must satisfy the same identity against rank-one polynomials (Dantas et al., 8 Jul 2025). The general equivalence between linear and polynomial Daugavet properties shows that no gap remains at the level of spaces, but a possible gap at the level of individual non-linear centers has not been eliminated.

In its present form, the theory identifies polynomial Daugavet centers as the polynomial realization of the Daugavet equation, ties them to weak polynomial topology, and places them inside a tensorial framework where homogeneous polynomials, symmetric projective tensor products, and diameter-two geometry become different expressions of the same phenomenon (Zoca et al., 2019, Dantas et al., 8 Jul 2025).