# Two remarks on diameter 2 properties/Kaks markust diameeter-2 omaduste kohta.

1. 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.sup.*], [B.sub.X] is the closed unit ball of X, and [S.sub.X] stands for the unit sphere of X. By a slice of [B.sub.X] we mean a set of the form

S([x.sup.*], [alpha]) = {x [member of] [B.sub.X]: [x.sup.*] (x) > 1 - a},

where [x.sup.*] [member of] [S.sub.X*] and a > 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.sub.X] has diameter 2; and X has the strong diameter 2 property if every convex combination of slices of [B.sub.X] has diameter 2, i.e., the diameter of [[summation].sup.n.sub.n=1] [[lambda].sub.i][S.sub.i] is 2, whenever n [member of] N, [[lambda].sub.1], ..., [[lambda].sub.n] [greater than or equal to] 0, with [[summation].sup.n.sub.n=1] [[lambda].sub.i] = 1, and [S.sub.1], ..., [S.sub.n] are slices of [B.sub.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 p. 26]), which asserts that every nonempty relatively weakly open subset of [B.sub.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 [l.sub.p]-sum for 1 < p < [infinity] (cf. Theorem 1). On the other hand, the diameter 2 property is stable under [l.sub.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 = [infinity]) and Proposition 4.6 from [1].

2. STRONG DIAMETER 2 PROPERTY IS NEVER STABLE UNDER [l.sub.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 [l.sub.p]-sums for 1 < p < [infinity] (see [1, Theorem 3.2]). The same result is true, and even easier also, for p = 1 and p = [infinity]. For p = [infinity], the diameter 2 case was obtained by Lopez Perez ([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 [l.sub.p]-sums (see ([1, Question (c)]). The answer was known for p = 1 and for p = [infinity]:

* If the Banach spaces X and Y have the strong diameter 2 property, then X [[direct sum].sub.1] Y has the strong diameter 2 property (see [1, Theorem 2.7 (iii)]). This result is essentially due to Becerra Guerrero and Lopez Perez in [4, proof of Lemma 2.1 (ii)].

* If a Banach space X has the strong diameter 2 property, then X [[direct sum].sub.[infinity]] Y has the strong diameter 2 property for any Banach space Y ([1, Proposition 4.6]). 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.

Theorem 1. Let X and Y be nontrivial Banach spaces and let 1 < p < [infinity] The Banach space Z = X [[direct sum].sub.p] Y fails the strong diameter 2 property.

Remark.

(1) Theorem 1 is a joint result with Mart Poldvere.

(2) Theorem 1 was obtained independently by Maria Acosta, Julio Becerra Guerrero, and Gines Lopez Perez; it is included in [2, Theorem 3.2].

(3) Eve Oja has presented another proof of Theorem 1 ([8]).

To prove Theorem 1, we will need the following elementary lemma.

Lemma 2. Let 1 < p < [infinity] and let q be such that 1/p + 1/q = 1. If [z.sup.*] = ([x.sup.*], [y.sup.*]) is an element in [S.sub.Z*] = [S.sub.X*][[direct sum].sub.q]Y*, then for every [member of] > 0 there exists a > 0 such that

[[parallel]([parallel]x[parallel], [parallel]y[parallel]) - ([[parallel][x.sup.*][parallel].sup.q-1], [[parallel][y.sup.*] [parallel].sup.q-1]) [parallel].sub.p] < e,

whenever z = (x, y) is an element in S([z.sup.*], [alpha]).

Proof. Note that if z = (x, y) is an element in S([z.sup.*], [alpha]), then ([parallel]x[parallel], [parallel]y[parallel]) and ([[parallel][x.sup.*][parallel].sup.q-1], [[parallel][y.sup.*][parallel].sup.q-1]) are both elements of the slice [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Obviously, when a tends to 0, then diam(S(([parallel][x.sup.] [parallel], [parallel][y.sup.*][parallel]), a)) tends to 0 as well. This proves the result. []

Proof of Theorem 1. In fact, we will show a stronger statement: For every [lambda] [member of] (0,1), there exists [alpha], [beta] > 0 and [z.sup.*], [[??].sup.*] [member of] [S.sub.Z*] such that

[lambda]S([z.sup.*], [alpha]) + (1 - [lambda])S([[??].sup.*], [alpha]) [subset] (1 - [beta])[B.aub.z].

Let [z.sup.*] [member of] [S.sub.X*] and [y.sup.*] [member of] [S.sub.Y*]. We take [z.sup.*] = ([x.sup.*], 0) and [[??].sup.*] = (0, [y.sup.*]). Then [z.sup.*] and [[??].sup.*] are elements in [S.sub.z*]. Fix [lambda] [member of] (0,1). Let

[epsilon] = 1 - [([[lambda].sup.p] + (1 - [lambda]).sup.p]).sup.1/p].

Clearly, [epsilon] > 0. By Lemma 2, there exists [alpha] > 0 such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

whenever z = (x, y) [member of] S([z.sup.*], [alpha]) and [??] = ([??],[??]) [member of] S([[??].sup.*], [alpha]).

One may take [beta] = [epsilon]/2. Indeed, for z = (x,y) [member of] S([z.sup.*], a) and [??] = ([??], [??]) [member of] S([[??].sup.*], [alpha]), we now have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] []

3. DIAMETER 2 PROPERTIES CARRY OVER TO THE WHOLE SPACE FROM A NONZERO M-IDEAL

We denote the annihilator of a subspace Y of a Banach space X by

[Y.sup.[perpendicular]] = {[x.sup.*] [member of] [X.sup.*]: [x.sup.*](y) = 0 for all y [member of] Y}.

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.sup.[perpendicular]] and

[parallel][x.sup.*][parallel] = [parallel][Px.sup.*][parallel] + [parallel][x.sup.*] - [Px.sup.*][parallel] for all [x.sup.*] [member of] [X.sup.*].

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 [[direct sum].sub.[infinity]] [x.sub.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 = [infinity]) 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 m [[summation].sup.n.sub.n=1] [[lambda].sub.i]S([x.sup.*.sub.i], [[alpha].sub.i]) be a convex combination of slices of [B.sub.X], where n [member of] N, and [[lambda].sub.1], ..., [[lambda].sub.n] [greater than or equal to] 0 such that [[summation].sup.n.sub.n=1] [[lambda].sub.i] = 1. Let [epsilon] > 0 be such that [epsilon] < min{[[alpha].sub.1], ..., [[alpha].sub.n]}/3.

We will show the existence of [x.sup.1.sub.1], ..., [x.sup.1.sub.n], ..., [x.sup.2.sub.n] [member of] [B.sub.X] such that [x.sup.k.sub.i] [member of] S([x.sup.*.sub.i] [[alpha].sub.i]) for every i = 1, ..., n, k = 1, 2, and

[parallel][m.summation over (i=1)] [[lambda].sub.i]([x.sup.1.sub.i] - [x.sup.2.sub.i])[parallel] > 2 - [epsilon]/1 + [epsilon].

Denote by P the M-ideal projection on X* with ker P = [Y.sup.[perpendicular]]. For every i = 1, ..., n, we take

[y.sup.*.sub.i] = [Px.sup.*.sub.i]/[parallel][Px.sup.*.sub.i][parallel] and [[beta].sub.i] = [epsilon] - [epsilon] [parallel][Px.sup.*.sub.i][parallel] + [[epsilon].sup.2]/[parallel][Px.sup.*.sub.i][parallel].

Note that, if [Px.sup.*.sub.i] [not equal to] 0, then [[beta].sub.i] > 0. If [Px.sup.*.sub.i] = 0, we can take [y.sup.*.sub.i] [member of] [S.sub.Y*] and[[beta].sub.i] > 0 to be arbitrary. Observe that [[summation].sup.n.sub.n=1][[lambda].sub.1]S([y.sup.*.sub.i], [[beta].sub.i]) is a convex combination of slices of [B.sub.Y]. Since Y has the strong diameter 2 property, we can find [y.sup.1.sub.1], ..., [y.sup.1.sub.n] and [y.sup.2.sub.n], ..., [y.sup.2.sub.n] in [B.sub.Y] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

There are [x.sub.1], ..., [x.sub.n] [member of] [B.sub.X] such that

([x.sup.*.sub.i] - [Px.sup.*.sub.i])([x.sub.i]) > ([parallel] [x.sup.*.sub.i] - [Px.sup.*.sub.i][parallel] - [epsilon])(1 + [epsilon])

for every i = 1, ..., n.

Since Y is an M-ideal in X, then by [11, Proposition 2.3], we can, for every i = 1, ..., n, choose [z.sub.i] [member of] [B.sub.Y] such that

[parallel][y.sup.k.sub.i] + [x.sub.i] - [z.sub.i] [parallel] < 1 + [epsilon], k = 1,2,

and

[absolute value of [Px.sup.*.sub.i] ([x.sub.i] - [z.sub.i])] < [epsilon].

We take [x.sup.k.sub.i] = [y.sup.k.sub.i] + [x.sub.i] - [z.sub.i]/1 + [epsilon], k = 1, 2, i = 1, ..., n.

Now, for every i = 1, ..., n, for every = 1, 2, [x.sup.k.sub.i] is an element in S([x.sup.*.sub.i], [[alpha].sub.i]), because

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Finally, observe that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. []

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 ofX. 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.sub.X] containing an element x0. We may assume that

{x [member of] [B.sub.X]: [absolute value of [x.sup.*.sub.i] (x - [x.sub.0])] < [gamma], i = 1, ..., n} [subset] U,

for some n [member of] N, [x.sup.*.sub.1], ..., [x.sup.*.sub.i] [member of] [S.sub.X*], and [gamma] > 0.

Denote by P the M-ideal projection on X* with ker P = [Y.sup.[perpendicular]], and let [delta] = max {[parallel] [Px.sup.*.sub.i][parallel] : i = 1, ..., n}. Let [epsilon] > 0 be such that [epsilon](4 + [delta]) < [gamma] We will show the existence of elements x and [??] in U such that

[parallel]x - [??][parallel] > 2- [epsilon]/1 + [epsilon].

Since [B.sub.Y] is dense in [B.sub.X] in the weak topology [sigma](X, ran P), we can find an element [y.sub.0] [member of] [B.sub.Y] such that

[absolute value of [Px.sup.*.sub.i]] ([x.sub.0] - [y.sub.0])] < [epsilon]

for every i = 1, ..., n. Consider the set

V = {y [member of] [B.sub.Y]: [absolute value of [Px.sup.*.sub.i]] (y - [y.sub.0])] < [epsilon] ([delta] + 1), i = 1, ..., n}.

Clearly V is a nonempty relatively weakly open subset of [B.sub.Y]. By the assumption, there are [y.sub.1], [y.sub.2] [member of] V with [parallel][y.sub.1] - [y.sub.2][parallel] > 2 - [epsilon].

Since Y is an M-ideal in X, by [11, Proposition 2.3], there is an element [z.sub.0] [member of] [B.sub.Y] such that

[parallel][y.sub.k] - [x.sub.0] - [z.sub.0][parallel] < 1 + [epsilon], k = 1, 2,

and

[absolute value of [Px.sup.*.sub.i]] ([x.sub.0] - [z.sub.0])] < [epsilon]

for every i = 1, ..., n.

We take

[x.sub.1] = [y.sub.1] + [x.sub.0] - [z.sub.0]/1 + [epsilon] and [x.sub.2] = [y.sub.2] + [x.sub.0] - [z.sub.0]/1 + [epsilon].

Now, for every i = 1, ..., n, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus, [x.sub.1] [member of] U. Similarly one can show that [x.sub.2] [member of] U. Finally, observe that

[parallel][x.sub.1] - [x.sub.2][parallel] = 1/1 + [epsilon] [parallel][y.sub.1] - [y.sub.2][parallel] > 2 - [epsilon]/1 + [epsilon]

doi: 10.3176/proc.2014.1.02

ACKNOWLEDGEMENTS

We would like to express our gratitude to Mart Poldvere and Olav Nygaard for fruitful conversations about the subject of this paper. We are grateful to Maria Acosta, Julio Becerra Guerrero, and Gines Lopez Perez for sending us their preprint [2], where the difference of the strong diameter 2 property and the diameter 2 property is obtained independently. Our recognition in [2] is very much appreciated. We are thankful to Eve Oja for useful comments concerning the presentation of this note and to the referees for helpful comments that improved our paper and for pointing out errors in our manuscript. This research was supported by Estonian Targeted Financing Project SF0180039s08 and by Estonian Science Foundation Grant 8976.

REFERENCES

[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 Lopez Perez, G. Stability results of diameter two properties. J. Conv. Anal. (to appear).

[3.] Acosta, M. D., Becerra Guerrero, J., and Rodriguez 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 Lopez Perez, 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 Rodriguez 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.] Lopez Perez, 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.

[11.] Werner, D. M-ideals and the "basic inequality". J. Approx. Theory, 1994, 76, 21-30.

Received 8 May 2013, revised 16 September 2013, accepted 23 September 2013, available online 14 March 2014

Rainis Haller * and Johann Langemets

Institute of Mathematics, University of Tartu, J. Liivi 2, 50409 Tartu, Estonia; johann.langemets@ut.ee

* Corresponding author, rainis.haller@ut.ee In this note, we summarize some main results on diameter 2 properties obtained in the Master's Thesis of the second named author.

The thesis was defended at the University of Tartu in June 2012.