Weakly open sets in the unit ball of some Banach spaces and the centralizer

We show that every Banach space X whose centralizer is infinite-dimensional satisfies that every nonempty weakly open set in BY has diameter 2, where Y = N,s,πX (N-fold symmetric projective tensor product of X, endowed with the symmetric projective norm), for every natural number N. We provide examples where the above conclusion holds that includes some spaces of operators and infinite-dimensional C∗-algebras. We also prove that every non-empty weak∗ open set in the unit ball of the space of Nhomogeneous and integral polynomials on X has diameter two, for every natural number N, whenever the Cunningham algebra of X is infinite-dimensional. Here we consider the space of N-homogeneous integral polynomials as the dual of the space N,s,εX (N-fold symmetric injective tensor product of X, endowed with the symmetric injective norm). For instance, every infinite-dimensional L1(μ) satisfies that its Cunningham algebra is infinite-dimensional. We obtain the same result for every non-reflexive L-embedded space, and so for every predual of an infinite-dimensional von Neumann algebra. © 2010 Elsevier Inc. All rights reserved.