Daugavet property in projective symmetric tensor products of Banach spaces Rueda Zoca, Abraham Martín Suárez, Miguel Daugavet property Polynomial Daugavet property Symmetric tensor product Projective tensor product L1-predual Miguel Martín partially supported by Spanish AEI Project PGC2018-093794-B- I00/AEI/10.13039/501100011033 (MCIU/AEI/FEDER, UE), A-FQM-484-UGR18 (Universidad de Granada and Junta de Analucía/FEDER, UE), FQM-185 (Junta de Andalucía/FEDER, UE), and by “Maria de Maeztu” Excellence Unit IMAG, reference CEX2020-001105-M funded by MCIN/AEI/10.13039/501100011033. Abraham Rueda Zoca was supported by Juan de la Cierva-Formación fellowship FJC2019-039973, by MTM2017-86182-P (Government of Spain, AEI/FEDER, EU), by Spanish AEI Project PGC2018-093794-B- I00/AEI/10.13039/501100011033 (MCIU/AEI/FEDER, UE), by Fundación Séneca, ACyT Región de Murcia grant 20797/PI/18, by Junta de Andalucía Grant A-FQM-484-UGR18 and by Junta de Andalucía Grant FQM-0185. 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. 2022-04-21T08:40:53Z 2022-04-21T08:40:53Z 2022-04-04 info:eu-repo/semantics/article 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] http://hdl.handle.net/10481/74424 10.1007/s43037-022-00186-6 eng http://creativecommons.org/licenses/by/3.0/es/ info:eu-repo/semantics/openAccess Atribución 3.0 España Birkhauser