Let A be a Banach algebra, and let D

Abstract. We prove that a finite-dimensional Banach space X has numerical index 0 if and only if it is the direct sum of a real space X 0 and nonzero complex spaces X 1 , . . . , Xn in such a way that the equalityholds for suitable positive integers q 1 , . . . , qn, and every ρ ∈ R and every x j ∈ X j (j = 0, 1, . . . , n). If the dimension of X is two, then the above result gives X = C, whereas dim(X) = 3 implies that X is an absolute sum of R and C. We also give an example showing that, in general, the number of complex spaces cannot be reduced to one.

A Banach space is said to have the diameter two property if every non-empty relatively weakly open subset of its unit ball has diameter two. We prove that the projective tensor product of two Banach spaces whose centralizer is infinite-dimensional has the diameter two property. The same statement also holds for X ⊗ π Y if the centralizer of X is infinitedimensional and the unit sphere of Y * contains an element of numerical index one.We provide examples of classical Banach spaces satisfying the assumptions of the results.If K is any infinite compact Hausdorff topological space, then C(K ) ⊗ π Y has the diameter two property for any nonzero Banach space Y . We also provide a result on the diameter two property for the injective tensor product.

A n.c.JB*-algebra is associative and commutative if and only if it has no non-zero nilpotent elements. This generalizes the analogous theorem of Kaplansky forC*-algebras. Different characterizations of associativity are given.

We prove that, if the centralizer of a Banach space X is infinite-dimensional, then every nonempty relatively weakly open subset of the closed unit ball of X has diameter equal to 2. This result, together with a suitable refinement also proven in the paper, contains (and improves in some cases) previously known facts for C * -algebras, J B * -triples, spaces of vector valued continuous functions, and spaces of operators.