A Banach space is said to have the diameter 2 property if the diameter of every nonempty relatively weakly open subset of its unit ball equals 2. In a paper by Abrahamsen, Lima, and Nygaard (Remarks on diameter 2 properties. J. Conv. Anal., 2013, 20, 439–452), the strong diameter 2 property is introduced and studied. This is the property that the diameter of every convex combination of slices of its unit ball equals 2. It is known that the diameter 2 property is stable by taking ℓp-sums for 1 ≤ p ≤ ∞. We show the absence of the strong diameter 2 property on ℓp-sums of Banach spaces when 1 < p < ∞. This confirms the conjecture of Abrahamsen, Lima, and Nygaard that the diameter 2 property and the strong diameter 2 property are different. We also show that the strong diameter 2 property carries over to the whole space from a non-zero M-ideal.
1. Abrahamsen, T., Lima, V., and Nygaard, O. Remarks on diameter 2 properties. J. Conv. Anal., 2013, 20, 439–452.
2. Acosta, M. D., Becerra Guerrero, J., and López Pérez, G. Stability results of diameter two properties. J. Conv. Anal. (to appear).
3. Acosta, M. D., Becerra Guerrero, J., and Rodríguez Palacios, A. Weakly open sets in the unit ball of the projective tensor product of Banach spaces. J. Math. Anal. Appl., 2011, 383, 461–473.
4. Becerra Guerrero, J. and López Pérez, G. Relatively weakly open subsets of the unit ball in functions spaces. J. Math. Anal. Appl., 2006, 315, 544–554.
5. Becerra Guerrero, J. and Rodríguez Palacios, A. Relatively weakly open sets in closed balls of Banach spaces, and the centralizer. Math. Z., 2009, 262, 557–570.
6. Ghoussoub, N., Godefroy, G., Maurey, B., and Schachermayer, W. Some topological and geometrical structures in Banach spaces. Mem. Amer. Math. Soc., 1987, 378, 1–116.
7. Harmand, P., Werner, D., and Werner, W. M-Ideals in Banach Spaces and Banach Algebras. Lecture Notes in Mathematics, vol. 1547, Springer, Berlin, 1993.
8. Oja, E. Sums of slices in direct sums of Banach spaces. Proc. Estonian Acad. Sci., 2014, 63, 8–10.
9. López Pérez, G. The big slice phenomena in M-embedded and L-embedded spaces. Proc. Amer. Math. Soc., 2005, 134, 273–282.
10. Nygaard, O. and Werner, D. Slices in the unit ball of a uniform algebra. Arch. Math., 2001, 76, 441–444.