Transfinite Daugavet property

TL;DR

This paper generalizes the Daugavet property to transfinite cardinals, characterizes these properties in function spaces, and explores their inheritance and applications in various Banach spaces.

Contribution

It introduces the transfinite Daugavet property, characterizes it in $C(K)$ spaces via a cardinal index, and studies its inheritance and applications in Banach space theory.

Findings

Characterization of transfinite Daugavet $C(K)$ spaces using a cardinal index.

The perfect Daugavet property relates to the absence of $G_\delta$-points.

Lipschitz space $ ext{Lip}(M)$ has the $oldsymbol{ extomega}$-perfect Daugavet property.

Abstract

We extend the Daugavet property and a perfect version of it to transfinite cardinals in order to distinguish between spaces with the ordinary Daugavet property by some kind of complexity (topological, density\ldots), providing a number of examples and results. First, we characterise the transfinite Daugavet $C(K)$ spaces in terms of a cardinal index $\mathfrak r(K)$, which generalises the notion of the reaping number of a Boolean algebra. Besides, the perfect Daugavet property characterizes the absence of $G_\delta$-points in $K$. We also study several inheritance results of the transfinite Daugavet properties by almost isometric ideals, absolute sums, and tensor product spaces, with a number of applications. We classify these properties for $L_1(\mu)$ and $L_\infty(\mu)$ spaces in terms of the Maharam's decomposition of the measure. We also show that the space of Lipschitz functions $\Lip(M)$ on a complete length metric space has the $\omega$-perfect Daugavet property, improving the previous knowledge.