Daugavet property and separability in Banach spaces. (English) Zbl 1391.46017

Rueda Zoca, Abraham

Banach J. Math. Anal. 12, No. 1, 68-84 (2018).

A Banach space \(X\) has the Daugavet property if \(\|I + T\| = 1 + \|T\|\) for every bounded linear operator \(T: X \to X\) of rank 1. The author demonstrates that a separable space \(X\) enjoys the Daugavet property if and only if, for every nonempty relatively weakly-star open subset \(W\) of \(B_{X^{**}}\), there exists \(u \in S_{X^{**}} \cap W\) such that \(\|x + u\| = 1 + \|x\|\) holds for every \(x \in X\).
A Banach space \(X\) is said to be \(L\)-embedded if there is a subspace \(Z \subset X^{**}\) such that \(X^{**} = X \oplus_1 Z\). It is demonstrated that a separable \(L\)-embedded space \(X\) possesses the Daugavet property if and only if the unit ball \(B_Z\) of the corresponding subspace \(Z\) is weak-star dense in \(B_{X^{**}}\). The main result of the paper says that, if \(X\) is a separable \(L\)-embedded Banach space with the Daugavet property, \(Y\) is a nonzero Banach space, and either \(X^{**}\) or \(Y\) has the metric approximation property, then the projective tensor product of \(X\) and \(Y\) has the Daugavet property.

Reviewer: Vladimir Kadets (Kharkiv)

MSC:

46B04	Isometric theory of Banach spaces
46B28	Spaces of operators; tensor products; approximation properties

Keywords:

Daugavet property; projective tensor product; separable Banach spaces; \(L\)-embedded Banach spaces; thickness index; JBW\(^*\)-triple