Extremal Structure of Projective Tensor Products García Lirola, Luis Rueda Zoca, Abraham Banach space Projective tensor product Preserved extreme point Strongly exposed point We 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 ⊗πY 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 ⊗πY (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 2 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 ⊗πX. 2023-09-05T07:47:35Z 2023-09-05T07:47:35Z 2023-07-31 journal article García-Lirola, L.C., Grelier, G., Martínez-Cervantes, G. et al. Extremal Structure of Projective Tensor Products. Results Math 78, 196 (2023). [https://doi.org/10.1007/s00025-023-01970-y] https://hdl.handle.net/10481/84240 10.1007/s00025-023-01970-y eng http://creativecommons.org/licenses/by/4.0/ open access Atribución 4.0 Internacional Springer Nature