Daugavet property in projective symmetric tensor products of Banach spaces
Metadata
Show full item recordEditorial
Birkhauser
Materia
Daugavet property Polynomial Daugavet property Symmetric tensor product Projective tensor product L1-predual
Date
2022-04-04Referencia bibliográfica
Martín, M., Rueda Zoca, A. Daugavet property in projective symmetric tensor products of Banach spaces. Banach J. Math. Anal. 16, 35 (2022). [https://doi.org/10.1007/s43037-022-00186-6]
Sponsorship
ACyT Región de Murcia 20797/PI/18; Universidad de Granada and Junta de Analucía; Fundación Séneca; European Commission; European Regional Development Fund; Junta de Andalucía FJC2019-039973, FQM-0185, MCIN/AEI/10.13039/501100011033, MTM2017-86182-P; University of the East FQM-185Abstract
We show that all the symmetric projective tensor products of a Banach space X
have the Daugavet property provided X has the Daugavet property and either X is an
L1-predual (i.e., X∗
is isometric to an L1-space) or X is a vector-valued L1-space. In
the process of proving it, we get a number of results of independent interest. For
instance, we characterise “localised” versions of the Daugavet property [i.e., Daugavet points and Δ-points introduced in Abrahamsen et al. (Proc Edinb Math Soc
63:475–496 2020)] for L1-preduals in terms of the extreme points of the topological dual, a result which allows to characterise a polyhedrality property of real
L1-preduals in terms of the absence of Δ-points and also to provide new examples of
L1-preduals having the convex diametral local diameter two property. These results
are also applied to nicely embedded Banach spaces [in the sense of Werner (J Funct
Anal 143:117–128, 1997)] so, in particular, to function algebras. Next, we show
that the Daugavet property and the polynomial Daugavet property are equivalent for
L1-preduals and for spaces of Lipschitz functions. Finally, an improvement of recent
results in Rueda Zoca (J Inst Math Jussieu 20(4):1409–1428, 2021) about the Daugavet property for projective tensor products is also obtained.