Banach spaces with small weakly open subsets of the unit ball and massive sets of Daugavet and $\Delta $-points

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2024年4月22日

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Christian Cobollo,
Daniel Isert,
Ginés López-Pérez,
Miguel Martín,
Yoël Perreau,
Alicia Quero,
Andrés Quilis,
Daniel L. Rodríguez-Vidanes,
Abraham Rueda Zoca

摘要

We prove that there exists an equivalent norm $\left| \left| \left| \cdot \right| \right| \right| $ $|||\cdot|||$ on $L_\infty [0,1]$ $L_{\infty} [0, 1]$ with the following properties: (1)

The unit ball of $(L_\infty [0,1],\left| \left| \left| \cdot \right| \right| \right| )$ $(L_{\infty} [0, 1], |||\cdot|||)$ contains non-empty relatively weakly open subsets of arbitrarily small diameter;

(2)

The set of Daugavet points of the unit ball of $(L_\infty [0,1],\left| \left| \left| \cdot \right| \right| \right| )$ $(L_{\infty} [0, 1], |||\cdot|||)$ is weakly dense;

(3)

The set of ccw $\Delta $ $Δ$ -points of the unit ball of $(L_\infty [0,1],\left| \left| \left| \cdot \right| \right| \right| )$ $(L_{\infty} [0, 1], |||\cdot|||)$ is norming.

(L_{\infty} [0, 1], |||\cdot|||)

which are not $\Delta $

Δ

(L_{\infty} [0, 1], |||\cdot|||)

(L_{\infty} [0, 1], |||\cdot|||)

provides both alternative and new examples that illustrate the differences between the various diametral notions for points of the unit ball of Banach spaces.

Banach spaces with small weakly open subsets of the unit ball and massive sets of Daugavet and \(\Delta \)-points

摘要