# Sums of slices in direct sums of Banach spaces/Viilude summad Banachi ruumide otsesummades.

Let X be a Banach space (over the real field R). Let BX and SX
denote its closed unit ball and its unit sphere, respectively. A slice
of [B.sub.X] is a set S([x.sup.*], [alpha]) := {x E [B.sub.X] :
[x.sup.*](x) > 1 - [alpha]}, where [MATHEMATICAL EXPRESSION NOT
REPRODUCIBLE IN ASCII] and [alpha] > 0.

The maximal diameter of slices can be 2. The same clearly holds for convex combinations of slices: diam ([[summation].sup.n.sub.(i=i)] [[lambda].sub.i] [S.sub.i]) [less than or equal to] 2 when [S.sub.i] are slices and [[lambda].sub.i] [grater than or equal to]0 satisfy ([[summation].sup.n.sub.(i=i)] [[lambda].sub.i] = 1. In studies on "big" slice phenomena, some recent attention has been paid to Banach spaces X where the diameter of every convex combination of slices equals 2 (see, e.g., [1,2,4] for results and references). Such spaces are said to have the strong diameter 2 property (for this and related terminology, see [1]). For instance, it is known that the direct sums [c.sub.0] [[direct sum].sub.1][c.sub.0] and [c.sub.0] [[direct sum].sub.[infinity]] [c.sub.0] have the strong diameter 2 property.

Answering (in the negative) a question posed in [1, Question (c)], R. Haller, J. Langemets, and M. Poldvere showed that the [l.sub.p]-sum [c.sub.0] [[direct sum].sub.p] [c.sub.0] fails the strong diameter 2 property whenever 1 < p < [infinity]. More generally, they proved the theorem (see [4, Theorem 1]): if X [not equal to] {0} and Y [not equal to] {0} are Banach spaces and 1 < p < [infinity], then X [[direct sum].sub.p] Y fails the strong diameter 2 property.

The same result was independently obtained in [2, Theorem 3.2] in a more general context of absolute norms. In both cases, among others, two slices [S.sub.1] and [S.sub.2] were exhibited, whose arithmetic mean 1/2([S.sub.1] + [S.sub.2]) had diameter strictly less than 2.

In this short note, we shall present a simple proof of the above result. Since we also would like to expose the reason for the failure of the strong diameter 2 property in X [[direct sum].sub.p] Y, we shall consider the following general case of [direct sum] Y.

Definition. Let X and Y be Banach spaces. We say that X [direct product] Y is equipped with a uniformly convex [R.sup.2]-norm if [parallel](x,y)[parallel] = [absolute value of ([parallel]x[parallel],[parallel]y[parallel]] for all x [member of] X and y [member of] Y, where [absolute value of((.,.))] is a uniformly convex norm on [R.sup.2] such that [absolute value of ((1, 0))] = [absolute value of ((0,1))] = 1 and [absolute value of ((a,b)] [less than or equal to] [absolute value of ((c, d))] whenever 0 [less than or equal to] a [less than or equal to] c and 0 [less than or equal to] b [less than or equal to] d.

Recall (see, e.g., [5, pp. 59-60]) that a norm [parallel]x[parallel] of a Banach space X is uniformly convex if its modulus of convexity [delta]X : (0,2] [right arrow] [0,1], defined by

[[delta].sub.X]([epsilon]) = inf {1 [[parallel] x + y [parallel]/2]: x, y [member of] X, [parallel] x [parallel], [parallel] y [parallel] [less than or equal to] 1, [parallel]x - y[parallel] [greater than or equal to] [epsilon]},

is strictly positive (i.e., [[delta].sub.X]([epsilon] > 0 when [epsilon] > 0).

The modulus of convexity measures (as nicely expressed in [3, p. 145]) the minimum depth of midpoints of line segments [x, y] in [B.sub.X] below the unit sphere [S.sub.X]. If X is uniformly convex, then SX contains no (nontrivial) line segments, i.e., X is strictly convex. (The converse holds when X is of finite dimension.)

Let S be a non-empty subset of a Banach space X = (X, [parallel]x[parallel]). In the following theorem, we shall use the notation

[parallel]S[parallel] := sup {[parallel]x[parallel] : x [member of] S}.

Since S [subset] [parallel]S[parallel] [B.sub.X], we clearly have

diam [1/2] S [less than or equal to] [parallel]S[parallel].

This is a rough estimate: if S is a slice, then [parallel]S[parallel] = 1, but diam S can be arbitrarily small (e.g., when X = [l.sup.2.sub.p], 1 < p < [infinity]).

Theorem. Let X [not equal to] {0} and Y [not equal to] {0} be Banach spaces. If X [direct product] Y is equipped with a uniformly convex [R.sup.2]-norm, then there are slices [S.sub.1] and [S.sub.2] of [B.sub.X [direct sum] Y] such that

[parallel][S.sub.1] + [S.sub.2] [parallel] < 2.

Consequently,

diam [1/2]([S.sub.1] + [S.sub.2]) < 2,

and X [direct sum] Y fails the strong diameter 2 property.

Proof. Let 0 < [alpha] < 1. We choose a such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. This can be done, because if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all such [alpha], then (1,1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], meaning that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Hence, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] would contain the line segment [(1,0), (1,1)], which is a contradiction.

Denote [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Since [[sigma].sub.1] and [[sigma].sub.2] are disjoint non-empty compact subsets, one has [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. If [u.sub.i] [member of] [[sigma].sub.i], i = 1,2, then [absolute value of ([u.sub.1] - [u.sub.2])] [greater than or equal to] [epsilon] and therefore 1 [absolute value of ([u.sub.1] + [u.sub.2])] [greater than or equal to][delta]. Hence [absolute value of ([u.sub.1] + [u.sub.2])] [less than or equal to] 2 (1 - [delta] meaning that

[absolute value of ([[sigma].sub.1] + [[sigma].sub.2])] [less than or equal to] 2 (1 - [delta]).

Take now any [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Consider the slices [S.sub.1] = S(([x.sup.*], 0), [alpha]) and [S.sub.2] = S((0, [y.sup.*]), [alpha]) of [B.sub.X [direct product] Y]. We shall show that

[parallel][S.sub.1] + [S.sub.2][parallel] [less than or equal to] [absolute value of ([[sigma].sub.1] + [[sigma].sub.2])],

which would complete the proof of the theorem.

First, observe that if (x,y) [member of] [S.sub.i], then ([parallel]x[parallel], [parallel]y[parallel]) [member of] [[sigma].sub.i], i = 1,2. Indeed, [absolute value of ([parallel]x[parallel], [parallel]y[parallel]))] [less than or equal to] 1 and, e.g.,

[parallel]x[parallel] [greater than or equal to] [x.sup.*] = ([x.sup.*], 0) (x, y) > 1 - [alpha].

Hence, for all ([x.sub.i], [y.sub.i]) [member of] [S.sub.i], i = 1,2, denoting [u.sub.i] = ([parallel][x.sub.i] [parallel], [parallel][y.sub.i] [parallel]) [member of] [[sigma].sub.i], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

as desired.

doi: 10.3176/proc.2014.1.03

ACKNOWLEDGEMENTS

The author is grateful to the referees for the comments and to Maria Acosta, Julio Becerra Guerrero, and Gines Lopez Perez for sending her the preprint of [2]. 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.] Day, M. M. Normed Linear Spaces, Third Ed. Springer, Berlin-Heidelberg, 1973.

[4.] Haller, R. and Langemets, J. Two remarks on diameter 2 properties. Proc. Estonian Acad. Sci., 2014, 63, 2-7.

[5.] Lindenstrauss, J. and Tzafriri, L. Classical Banach Spaces II: Function Spaces. Springer, Berlin-Heidelberg, 1979.

Received 13 May 2013, accepted 10 June 2013, available online 14 March 2014

Eve Oja

Institute of Mathematics, University of Tartu, J. Liivi 2, 50409 Tartu, Estonia

Estonian Academy of Sciences, Kohtu 6, 10130 Tallinn, Estonia; eve.oja@ut.ee

The maximal diameter of slices can be 2. The same clearly holds for convex combinations of slices: diam ([[summation].sup.n.sub.(i=i)] [[lambda].sub.i] [S.sub.i]) [less than or equal to] 2 when [S.sub.i] are slices and [[lambda].sub.i] [grater than or equal to]0 satisfy ([[summation].sup.n.sub.(i=i)] [[lambda].sub.i] = 1. In studies on "big" slice phenomena, some recent attention has been paid to Banach spaces X where the diameter of every convex combination of slices equals 2 (see, e.g., [1,2,4] for results and references). Such spaces are said to have the strong diameter 2 property (for this and related terminology, see [1]). For instance, it is known that the direct sums [c.sub.0] [[direct sum].sub.1][c.sub.0] and [c.sub.0] [[direct sum].sub.[infinity]] [c.sub.0] have the strong diameter 2 property.

Answering (in the negative) a question posed in [1, Question (c)], R. Haller, J. Langemets, and M. Poldvere showed that the [l.sub.p]-sum [c.sub.0] [[direct sum].sub.p] [c.sub.0] fails the strong diameter 2 property whenever 1 < p < [infinity]. More generally, they proved the theorem (see [4, Theorem 1]): if X [not equal to] {0} and Y [not equal to] {0} are Banach spaces and 1 < p < [infinity], then X [[direct sum].sub.p] Y fails the strong diameter 2 property.

The same result was independently obtained in [2, Theorem 3.2] in a more general context of absolute norms. In both cases, among others, two slices [S.sub.1] and [S.sub.2] were exhibited, whose arithmetic mean 1/2([S.sub.1] + [S.sub.2]) had diameter strictly less than 2.

In this short note, we shall present a simple proof of the above result. Since we also would like to expose the reason for the failure of the strong diameter 2 property in X [[direct sum].sub.p] Y, we shall consider the following general case of [direct sum] Y.

Definition. Let X and Y be Banach spaces. We say that X [direct product] Y is equipped with a uniformly convex [R.sup.2]-norm if [parallel](x,y)[parallel] = [absolute value of ([parallel]x[parallel],[parallel]y[parallel]] for all x [member of] X and y [member of] Y, where [absolute value of((.,.))] is a uniformly convex norm on [R.sup.2] such that [absolute value of ((1, 0))] = [absolute value of ((0,1))] = 1 and [absolute value of ((a,b)] [less than or equal to] [absolute value of ((c, d))] whenever 0 [less than or equal to] a [less than or equal to] c and 0 [less than or equal to] b [less than or equal to] d.

Recall (see, e.g., [5, pp. 59-60]) that a norm [parallel]x[parallel] of a Banach space X is uniformly convex if its modulus of convexity [delta]X : (0,2] [right arrow] [0,1], defined by

[[delta].sub.X]([epsilon]) = inf {1 [[parallel] x + y [parallel]/2]: x, y [member of] X, [parallel] x [parallel], [parallel] y [parallel] [less than or equal to] 1, [parallel]x - y[parallel] [greater than or equal to] [epsilon]},

is strictly positive (i.e., [[delta].sub.X]([epsilon] > 0 when [epsilon] > 0).

The modulus of convexity measures (as nicely expressed in [3, p. 145]) the minimum depth of midpoints of line segments [x, y] in [B.sub.X] below the unit sphere [S.sub.X]. If X is uniformly convex, then SX contains no (nontrivial) line segments, i.e., X is strictly convex. (The converse holds when X is of finite dimension.)

Let S be a non-empty subset of a Banach space X = (X, [parallel]x[parallel]). In the following theorem, we shall use the notation

[parallel]S[parallel] := sup {[parallel]x[parallel] : x [member of] S}.

Since S [subset] [parallel]S[parallel] [B.sub.X], we clearly have

diam [1/2] S [less than or equal to] [parallel]S[parallel].

This is a rough estimate: if S is a slice, then [parallel]S[parallel] = 1, but diam S can be arbitrarily small (e.g., when X = [l.sup.2.sub.p], 1 < p < [infinity]).

Theorem. Let X [not equal to] {0} and Y [not equal to] {0} be Banach spaces. If X [direct product] Y is equipped with a uniformly convex [R.sup.2]-norm, then there are slices [S.sub.1] and [S.sub.2] of [B.sub.X [direct sum] Y] such that

[parallel][S.sub.1] + [S.sub.2] [parallel] < 2.

Consequently,

diam [1/2]([S.sub.1] + [S.sub.2]) < 2,

and X [direct sum] Y fails the strong diameter 2 property.

Proof. Let 0 < [alpha] < 1. We choose a such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. This can be done, because if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all such [alpha], then (1,1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], meaning that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Hence, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] would contain the line segment [(1,0), (1,1)], which is a contradiction.

Denote [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Since [[sigma].sub.1] and [[sigma].sub.2] are disjoint non-empty compact subsets, one has [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. If [u.sub.i] [member of] [[sigma].sub.i], i = 1,2, then [absolute value of ([u.sub.1] - [u.sub.2])] [greater than or equal to] [epsilon] and therefore 1 [absolute value of ([u.sub.1] + [u.sub.2])] [greater than or equal to][delta]. Hence [absolute value of ([u.sub.1] + [u.sub.2])] [less than or equal to] 2 (1 - [delta] meaning that

[absolute value of ([[sigma].sub.1] + [[sigma].sub.2])] [less than or equal to] 2 (1 - [delta]).

Take now any [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Consider the slices [S.sub.1] = S(([x.sup.*], 0), [alpha]) and [S.sub.2] = S((0, [y.sup.*]), [alpha]) of [B.sub.X [direct product] Y]. We shall show that

[parallel][S.sub.1] + [S.sub.2][parallel] [less than or equal to] [absolute value of ([[sigma].sub.1] + [[sigma].sub.2])],

which would complete the proof of the theorem.

First, observe that if (x,y) [member of] [S.sub.i], then ([parallel]x[parallel], [parallel]y[parallel]) [member of] [[sigma].sub.i], i = 1,2. Indeed, [absolute value of ([parallel]x[parallel], [parallel]y[parallel]))] [less than or equal to] 1 and, e.g.,

[parallel]x[parallel] [greater than or equal to] [x.sup.*] = ([x.sup.*], 0) (x, y) > 1 - [alpha].

Hence, for all ([x.sub.i], [y.sub.i]) [member of] [S.sub.i], i = 1,2, denoting [u.sub.i] = ([parallel][x.sub.i] [parallel], [parallel][y.sub.i] [parallel]) [member of] [[sigma].sub.i], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

as desired.

doi: 10.3176/proc.2014.1.03

ACKNOWLEDGEMENTS

The author is grateful to the referees for the comments and to Maria Acosta, Julio Becerra Guerrero, and Gines Lopez Perez for sending her the preprint of [2]. 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.] Day, M. M. Normed Linear Spaces, Third Ed. Springer, Berlin-Heidelberg, 1973.

[4.] Haller, R. and Langemets, J. Two remarks on diameter 2 properties. Proc. Estonian Acad. Sci., 2014, 63, 2-7.

[5.] Lindenstrauss, J. and Tzafriri, L. Classical Banach Spaces II: Function Spaces. Springer, Berlin-Heidelberg, 1979.

Received 13 May 2013, accepted 10 June 2013, available online 14 March 2014

Eve Oja

Institute of Mathematics, University of Tartu, J. Liivi 2, 50409 Tartu, Estonia

Estonian Academy of Sciences, Kohtu 6, 10130 Tallinn, Estonia; eve.oja@ut.ee

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Title Annotation: | MATHEMATICS |
---|---|

Author: | Oja, Eve |

Publication: | Proceedings of the Estonian Academy of Sciences |

Article Type: | Report |

Date: | Mar 1, 2014 |

Words: | 1396 |

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