Transfinite 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.