Two remarks on diameter 2 properties

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 `p-sums for 1 ≤ p ≤ ∞. We show the absence of the strong diameter 2 property on `p-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.


INTRODUCTION
All Banach spaces considered in this note are over the real field.For a Banach space X, its dual space is denoted by X * , B X is the closed unit ball of X, and S X stands for the unit sphere of X.By a slice of B X we mean a set of the form S(x * , α) = {x ∈ B X : x * (x) > 1 − α}, where x * ∈ S X * and α > 0. Nygaard and Werner [10] showed that in every infinite-dimensional uniform algebra, every nonempty relatively weakly open subset of its closed unit ball has diameter 2. If a Banach space satisfies this condition, then it is said to have the diameter 2 property (see, e.g., [1,3,5]).
In addition to the diameter 2 property, Abrahamsen, Lima, and Nygaard [1] consider two other formally different diameter 2 properties -the local diameter 2 property and the strong diameter 2 property.
According to the terminology in [1], a Banach space X has the local diameter 2 property if every slice of B X has diameter 2; and X has the strong diameter 2 property if every convex combination of slices of B X has diameter 2, i.e., the diameter of ∑ n i=1 λ i S i is 2, whenever n ∈ N, λ 1 , . . ., λ n ≥ 0, with ∑ n i=1 λ i = 1, and S 1 , . . ., S n are slices of B X .
The diameter 2 property clearly implies the local diameter 2 property.The strong diameter 2 property implies the diameter 2 property.This follows directly from Bourgain's lemma ([6, Lemma II.1 p. 26]), which asserts that every nonempty relatively weakly open subset of B X contains some convex combination of slices.
It is conjectured in [1] that these three diameter 2 properties are different.In Section 2, we will show that there exist Banach spaces with the diameter 2 property but without the strong diameter 2 property.In fact, we prove that the strong diameter 2 property is never stable by taking the p -sum for 1 < p < ∞ (cf.Theorem 1).On the other hand, the diameter 2 property is stable under p -sums (see [1,Theorem 3.2]).
The papers [1] and [9] inspired us to consider diameter 2 properties in the context of M-ideals.Section 3 is the result of that study.We show that all three diameter 2 properties carry over to the whole space from a non-zero M-ideal.This generalizes Theorem 3.2 (the case of p = ∞) and Proposition 4.6 from [1].

STRONG DIAMETER 2 PROPERTY IS NEVER STABLE UNDER p -SUMS
Perhaps the most surprising result in [1] is that the local diameter 2 property and the diameter 2 property are stable by taking p -sums for 1 < p < ∞ (see [1,Theorem 3.2]).The same result is true, and even easier also, for p = 1 and p = ∞.For p = ∞, the diameter 2 case was obtained by López Pérez ([9, Lemma 2.1], see also [4,Lemma 2.2]).
One of the questions asked in [1] was whether the strong diameter 2 property is also stable under p -sums (see ([1, Question (c)]).The answer was known for p = 1 and for p = ∞: ).We will generalize the last result in Proposition 3.
The following is our main result.It provides an answer, in the negative, to Question (c) in [1].Moreover, it confirms the conjecture in [1] that the diameter 2 property and the strong diameter 2 property are different.To prove Theorem 1, we will need the following elementary lemma.

DIAMETER 2 PROPERTIES CARRY OVER TO THE WHOLE SPACE FROM A NON-ZERO M-IDEAL
We denote the annihilator of a subspace Y of a Banach space X by According to the terminology in [7], a closed subspace Y of a Banach space X is called an M-ideal if there exists a norm-1 projection P on X * with ker P = Y ⊥ and Relations between M-ideal structure and the diameter 2 property were first considered in [9].There it is proved that if a proper subspace Y of X is an M-ideal in X and the range of the corresponding projection is 1-norming, then both Y and X have the diameter 2 property (see [9,Theorem 2.4]).In [1, Theorem 4.10] it is shown that, under the same assumptions, one can conclude that both Y and X have even the strong diameter 2 property.An immediate corollary of this is that if a nonreflexive Banach space X is an M-ideal in its bidual, then both X and X * * have the strong diameter 2 property.
One cannot omit the assumption that the range of the corresponding projection is 1-norming.To see an example of this, let Y be any Banach space and let X = Y ⊕ ∞ c 0 .Then, by [1, Proposition 4.6] (or Proposition 3 below), X has the strong diameter 2 property and Y is an M-ideal in X.
In the following we will show that if a non-zero M-ideal Y has some diameter 2 property, then X has the same diameter 2 property without the assumption that the range of the projection is 1-norming.This, at the same time, generalizes Theorem 3.2 (the case of p = ∞) and the above-mentioned Proposition 4.6 of [1].Proposition 3. Let X be a Banach space and let Y be a proper closed subspace of X. Assume that Y is an M-ideal in X.If Y has the strong diameter 2 property, then X has the strong diameter 2 property.
Proof.Let ∑ n i=1 λ i S(x * i , α i ) be a convex combination of slices of B X , where n ∈ N, and λ 1 , . . ., λ n ≥ 0 such that ∑ n i=1 λ i = 1.Let ε > 0 be such that ε < min{α 1 , . . ., α n }/3.We will show the existence of x 1  1 , . . ., Denote by P the M-ideal projection on X * with ker P = Y ⊥ .For every i = 1, . . ., n, we take There are x 1 , . . ., x n ∈ B X such that Since Y is an M-ideal in X, then by [11, Proposition 2.3], we can, for every i = 1, . . ., n, choose and Now, for every i = 1, . . ., n, for every k = 1, 2, x k i is an element in S(x * i , α i ), because Finally, observe that We conclude our study with the local diameter 2 and the diameter 2 versions of Proposition 3.
Proposition 4. Let X be a Banach space and let Y be a proper closed subspace of X. Assume that Y is an M-ideal in X.If Y has the local diameter 2 property, then X has the local diameter 2 property.
Proof.Take n = 1 in the proof of Proposition 3.
The next result is obtained in the proof of [9, Theorem 2.4], but not stated explicitly.We will give a direct proof of this result.Proposition 5. Let X be a Banach space and let Y be a proper closed subspace of X. Assume that Y is an M-ideal in X.If Y has the diameter 2 property, then X has the diameter 2 property.
Proof.The proof is similar to the proof of Proposition 3.
Let U be a nonempty relatively weakly open subset of B X containing an element x 0 .We may assume that {x ∈ B X : , and γ > 0. Denote by P the M-ideal projection on X * with ker P = Y ⊥ , and let δ = max{ Px * i : i = 1, . . ., n}.Let ε > 0 be such that ε(4 + δ ) < γ.We will show the existence of elements x and x in U such that We take x 1 = y 1 + x 0 − z 0 1 + ε and x 2 = y 2 + x 0 − z 0 1 + ε .Now, for every i = 1, . . ., n, we have Thus, x 1 ∈ U. Similarly one can show that x 2 ∈ U. Finally, observe that

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If the Banach spaces X and Y have the strong diameter 2 property, then X ⊕ 1 Y has the strong diameter 2 property (see [1, Theorem 2.7 (iii)]).This result is essentially due to Becerra Guerrero and López Pérez in [4, proof of Lemma 2.1 (ii)].• If a Banach space X has the strong diameter 2 property, then X ⊕ ∞ Y has the strong diameter 2 property for any Banach space Y ([1, Proposition 4.6] we can take y * i ∈ S Y * and β i > 0 to be arbitrary.Observe that∑ n i=1 λ i S(y * i , β i ) is a convexcombination of slices of B Y .Since Y has the strong diameter 2 property, we can find y 1 1 , . . ., y 1 n and y 2 1 , . . ., y 2 n in B Y such that