doi:10.1007/s00025-023-01970-yengGarcía-Lirola, Luis C.Grelier, GuillaumeMartínez-Cervantes, GonzaloRueda Zoca, AbrahamExtremal Structure of Projective Tensor ProductsART-2023-135020We prove that, given two Banach spaces X and Y and bounded, closed convex sets C⊆X and D⊆Y , if a nonzero element z∈co¯¯¯¯¯¯(C⊗D)⊆X⊗ˆπY is a preserved extreme point then z=x0⊗y0 for some preserved extreme points x0∈C and y0∈D , whenever K(X,Y∗) separates points of X⊗ˆπY (in particular, whenever X or Y has the compact approximation property). Moreover, we prove that if x0∈C and y0∈D are weak-strongly exposed points then x0⊗y0 is weak-strongly exposed in co¯¯¯¯¯¯(C⊗D) whenever x0⊗y0 has a neighbourhood system for the weak topology defined by compact operators. Furthermore, we find a Banach space X isomorphic to ℓ2 with a weak-strongly exposed point x0∈BX such that x0⊗x0 is not a weak-strongly exposed point of the unit ball of X⊗ˆπX .2023http://zaguan.unizar.es/record/12791510.1007/s00025-023-01970-yhttp://zaguan.unizar.es/record/127915oai:zaguan.unizar.es:127915info:eu-repo/grantAgreement/ES/DGA/E48-23Rinfo:eu-repo/grantAgreement/ES/MICINN-AEI-FEDER/PID2021-122126NB-C31info:eu-repo/grantAgreement/ES/MICINN-AEI-FEDER/PID2021-122126NB-C32Results in Mathematics 78 (2023), 196 [16 pp.]byhttp://creativecommons.org/licenses/by/3.0/es/info:eu-repo/semantics/openAccess