Slicely countably determined Banach spaces

Abstract

Abstract. We introduce the class of slicely countably determined Banach spaces which contains in particular all spaces with the Radon-Nikodým property and all spaces without copies of 1 . We present many examples and several properties of this class. We give some applications to Banach spaces with the Daugavet and the alternative Daugavet properties, lush spaces and Banach spaces with numerical index 1. In particular, we show that the dual of a real infinite-dimensional Banach space with the alternative Daugavet property contains 1 and that operators which do not fix copies of 1 on a space with the alternative Daugavet property satisfy the alternative Daugavet equation.

Operations with slicely countably determined sets

“…In this section we prove two main results that answer [2,Question 7.5]. Namely, we demonstrate that the properties aSCD and SCD are not equivalent for general bounded closed convex sets, but in the most important case for the applications, namely that of balanced bounded closed convex sets, the equivalence holds true.…”

“…The property "slicely countably determined" (SCD for short) for Banach spaces and their subsets was first considered and studied in [1] (see [2] for the complete version), proving to have noticeable applications to Banach spaces with the Daugavet property, numerical index one and other related properties [2,7,9,10].…”

“…We can restrict ourselves to study bounded, closed and convex sets because of the following result ( [9,Proposition 7.20] and [2,Remark 2.7]). Since we are going to use it several times, we will sketch the proof for the readers' convenience.…”

“…Typical examples of spaces not possessing the Daugavet property but nevertheless having the alternative Daugavet property are c 0 and 1 . See[11] and[13].Theorem 4.4 of[2] says, in particular, that if X possesses the alternative Daugavet property and B X is SCD, then every bounded linear operator on X satisfies the alternative Daugavet equation. Theorem 5.3 and Proposition 5.8 of[2] say that if X possesses the (alternative) Daugavet property and T (B X ) is SCD, then the operator T satisfies the (alternative) Daugavet…”

Some Observations on the Numerical Index and the Polynomial Numerical Index

“…A. Avilés et al [1] generalized this fact for a strongly regular Banach space X. We recall that a closed convex bounded subset A of a Banach space is said to be strongly regular if every non-empty convex subset L of A contains a convex combination of slices of L of arbitrarily small diameter, and we say that X is strongly regular if B X is strongly regular.…”