PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 92(106) (2012), 139–143 DOI: 10.2298/PIM1206139H

ON THE EXTREME POINTS AND STRONGLY EXTREME POINTS IN KÖTHE–BOCHNER SPACES

Zhentao Hou and Jinghong Pan

Communicated by Stevan Pilipović

Abstract. We give the necessary conditions of extreme points and strongly extreme points in the unit ball of Köthe–Bochner spaces. The conditions have been shown to be sufficient earlier.

1. Introduction

Let (Ω, Σ, μ) denote a measure space with σ-finite and complete measure μ and L^{0} = L^{0}(Ω) denote the space of all (equivalence classes of) Σ-measurable real-valued functions, equipped with the topology of convergence in measure on μ-finite sets. In what follows, if x, y ∈ L^{0}, then x ⩽ y means x(t) ⩽ y(t) μ-almost everywhere in Ω. A Banach subspace E of L^{0} is said be a Köthe function space (over (Ω, Σ, μ)) if

(i) |x| ⩽ |y|, x ∈ L^{0}, y ∈ E imply x ∈ E and ∥x∥ ⩽ ∥y∥.

(ii) suppE := ⋃ {suppx : x ∈ E} = Ω, where suppx = {t ∈ Ω : x(t) ≠ 0}.

Köthe spaces are Banach lattices and detailed studies of Banach lattices and Köthe spaces may be found in [4]. A Köthe function space E is said to be order continuous provided x_{n} ↘ 0 implies ∥x_{n}∥ → 0. E is said to be strictly monotone if x ⩽ y and x ≠ y implies ∥x∥ < ∥y∥.

In this paper we always denote by E a Köthe function space on (Ω, Σ, μ), X a Banach space. By E(X) we denote the Banach space of all equivalence classes of strongly measurable function x : Ω → X such that x̃ = ∥x(·)∥_{X} ∈ E equipped with the norm ⫴x⫴ = ∥x̃∥_{E}.

The space (E(X), ⫴·⫴) is called the Köthe–Bochner space. The most important class of Köthe–Bochner function spaces E(X) are the Lebesgue–Bochner spaces L^{p}(X), (1 ⩽ p < ∞) and their generalization the Orlicz–Bochner spaces L^{ϕ}(X).

2010 Mathematics Subject Classification: Primary 46E30; Secondary 46B20.

Key words and phrases: Orlicz space, Köthe–Bochner space, extreme point, strongly extreme point.

The authors were supported by NSFC Grant #11171244.

They have been studied by many authors. The geometric properties of the Köthe– Bochner spaces have been also studied by many authors.

For any Banach space X, let S(X) and B(X) stand for the unit sphere and unit ball of X, respectively. An x ∈ S(X) is called an extreme point of B(X) if 2x = y+z, y, z ∈ B(X) implies y = z. Denote the set of all extreme points of B(X) by ExtB(X). An X is called rotund (R) provided that S(X) = ExtB(X). An x ∈ S(X) is said to be strongly extreme point of B(X) if for every sequence {x_{n}} in X, lim_{n} ∥x_{n} ± x∥ = 1 implies lim_{n} x_{n} = 0, or equivalently, for {x_{n}}, {y_{n}} ⊂ X, lim_{n} ∥x_{n}∥ = lim_{n} ∥y_{n}∥ = 1 and 2x = x_{n}+y_{n} (n = 1, 2, …) implies lim_{n} ∥x_{n}−y_{n}∥ =

0. An X is called mid-point locally uniformly rotund (MLUR) if the set of all strongly extreme points of B(X) is equal to S(X). A point x ∈ S(X) is said to be a point of local uniform rotundity (or a LUR-point) if for arbitrary sequence {x_{n}} ⊂ B(X), lim_{n} ∥x_{n} + x∥ = 2 and lim_{n} ∥x_{n}∥ = 1 imply lim_{n} x_{n} = x. A space X is locally uniformly uniformly rotund (LUR) if every point of S(X) is a LUR-point. Clearly, (LUR) ⇒ (MLUR) ⇒ (R).

In 2006, Ren, Feng and Wu [5] obtained the sufficient conditions for extreme points of B(E(X)). The sufficient or necessary conditions for strongly extreme points of B(E(X)) were discussed by Hudzik and Mastylo [2] and Cerdá, Hudzik, and Mastylo [1]. We list the results here:

Proposition 1.1. [5] Let E be a strictly monotone Köthe function space and X a real Banach space. Assume that x ∈ S(E(X)) have the following properties:

(a) x̃ = ∥x(·)∥_{X} is an extreme point of B(X).

(b) \frac{x(t)}{∥x(t)∥_{X}} is an extreme point of B(X) for a.e. t ∈ suppx.

Then x is an extreme point of B(E(X)).

Proposition 1.2. [2] Let E be a locally uniformly rotund Köthe function space and X a real Banach space. If x ∈ S(E(X)) is such that \frac{x(t)}{∥x(t)∥_{X}} is a strongly extreme point of B(X) for a.e. t ∈ suppx, then x is a strongly extreme point of B(E(X)).

Proposition 1.3. [2] Let E be as in Proposition 1.2 and X a separable Banach space. If x ∈ S(E(X)) is a strongly extreme point, then \frac{x(t)}{∥x(t)∥_{X}} is a strongly extreme point of B(X) for a.e. t ∈ suppx.

Proposition 1.4. [1] Let E be an order continuous Banach function lattice and X a real Banach space. Assume that x ∈ S(E(X)) have the following properties:

(a) x̃ = ∥x(·)∥_{X} is a strongly extreme point of B(X).

(b) \frac{x(t)}{∥x(t)∥_{X}} is a strongly extreme point of B(X) for a.e. t ∈ suppx.

Then x is a strongly extreme point of B(E(X)).

In this paper, we will show that the two properties in Proposition 1 and Proposition 4 are exactly the necessary conditions for x ∈ S(E(X)) to be extreme point and strongly extreme point of B(E(X)), respectively.

2. Main Results

Theorem 2.1. Let E be a Köthe function space and X a real Banach space. Assume that x ∈ S(E(X)) is an extreme point of B(E(X)). Then we have:

(a) x̃ = ∥x(·)∥_{X} is an extreme point of B(X).

(b) \frac{x(t)}{∥x(t)∥_{X}} is an extreme point of B(X) for a.e. t ∈ suppx.

Proof. (a) Suppose that ∥x(·)∥_{X} is not an extreme point of B(E). Then there exist two real measurable functions y(t), z(t), such that y(·), z(·) ∈ S(E) and 2∥x(t)∥_{X} = y(t) + z(t) but y ≠ z. Define

g(t) = \{ \begin{matrix}\frac{x(t)}{∥x(t)∥_{X}} , & t ∈ suppx, \\ x_{0}, & otherwise, \end{matrix}

where x_{0} is a fixed point in S(X). Obviously g(·) is measurable. Let f_{1}(t) = y(t)g(t), f_{2}(t) = z(t)g(t). Then ⫴f_{1}⫴ = ∥∥y(t)g(t)∥_{X} ∥_{E} = ∥y(t)∥_{E} = 1.

Similarly, |∥f_{2}∥| = 1. Moreover, f_{1}(t) + f_{2}(t) = (y(t) + z(t))g(t) = 2x(t) but f_{1} ≠ f_{2}. This contradicts the assumption that x is an extreme point.

(b) Observe that \frac{x(t)}{∥x(t)∥_{X}} ∈ S(X). Let A = \{t : \frac{x(t)}{∥x(t)∥_{X}} ⫋ ExtS(X)\}. Suppose

that μA > 0. Then there are y_{1}, z_{1} ∈ S(X) such that 2\frac{x(t)}{∥x(t)∥_{X}} = y_{1}(t) + z_{1}(t), ( ∈ A) but y_{1} ≠ z_{1}. Define

(y(t), z(t)) = \{ \begin{matrix}(x(t), x(t)), & t ∈ Ω ∖ A, \\ (∥x(t)∥_{X} y_{1}(t), ∥x(t)∥_{X} z_{1}(t)), & t ∈ A. \end{matrix}

Therefore, 2x(t) = y(t) + z(t) a.e. t ∈ Ω but z ≠ y. Moreover,

⫴y⫴ = ∥∥y(t)∥_{X} ∥_{E}
= ∥∥y(t)∥_{X} χ_{A} + ∥y(t)∥_{X} χ_{Ω∖A} ∥_{E}
= ∥∥∥x(t)∥_{X} y_{1}(t)∥_{X} χ_{A} + ∥x(t)∥_{X} χ_{Ω∖A} ∥_{E}
= ∥∥x(t)∥_{X} χ_{A} + ∥x(t)∥_{X} χ_{Ω\A} ∥ _{E}
= ∥∥x(t)∥_{X} ∥_{E} = ⫴x⫴ = 1.

Similarly, |∥z∥| = 1. This means that x is not an extreme point of B(E(X)), a contradiction. So μA = 0, which finishes the proof. □

Theorem 2.2. Let E be a Köthe function space and X a real Banach space. Assume that x is a strongly extreme point of B(E(X)). Then

(a) ∥x(·)∥_{X} is a strongly extreme point of B(E);

(b) \frac{x(t)}{∥x(t)∥_{X}} is a strongly extreme point of B(X) a.e. t ∈ suppx.

Proof. (a) Assume lim_{n→∞} ∥∥x(t)∥_{X} ±x_{n}(t)∥_{E} = 1, where {x_{n}} is a sequence of real valued functions. Define

g(t) = \{ \begin{matrix}\frac{x(t)}{∥x(t)∥_{X}} , & t ∈ suppx, \\ x_{0}, & otherwise, \end{matrix}

where x_{0} ∈ S(X) is a fixed point. Then g(t) is measurable. Moreover, ∥g(t)∥_{X} ≡ 1 and ∥x(t)∥_{X} g(t) ≡ x(t) for every t ∈ Ω. Observe that

1 = lim_{n→∞} ∥∥x(t)∥_{X} ± x_{n}(t)∥_{E}

= lim_{n→∞} |∥[∥x(t)∥_{X} ± x_{n}(t)]g(t)∥|

= lim_{n→∞} ⫴x(t) ± x_{n}(t)g(t) ⫴ .

Since x is a strongly extreme point in B(E(X)), we have

lim_{n→∞} ⫴x_{n}(t)g(t)⫴ = lim_{n→∞} ∥x_{n}(t)∥_{E} = 0

which implies that ∥x(·)∥_{X} is a strongly extreme point of B(E).

(b) Denote by z(t) = \frac{x(t)}{∥x(t)∥_{X}} . Define

A = {t ∈ suppx : z(t) is not a strongly extreme point of B(X)}.

We should prove μA = 0. Suppose it is not the case, e.g., μA > 0. Define

A_{mk} = { t ∈ A : there exists a sequence {z_{n}(t)} such that
1 − \frac{1}{n} < ∥z_{n}(t) ± z(t)∥_{X} < 1 + \frac{1}{n} , ∥z_{n}(t)∥_{X} ⩾ \frac{1}{m} for n ⩾ k\}.

Then ⋃ _{k=1}^{∞} ⋃ _{m=1}^{∞} A_{mk} = A. There are m_{0}, k_{0} ⩾ 1 such that μ(A_{m_{0} k_{0}} ) > 0. Let δ = \frac{1}{m_{0}} . We deduce that in the set B = A_{m_{0} k_{0}} , if n ⩾ k_{0}, then 1 − \frac{1}{n} < ∥z_{n}(t) ± z(t)∥_{X} < 1 + \frac{1}{n} and ∥z_{n}(t)∥_{X} ⩾ δ > 0.

However the functions z_{n}(·) need not be measurable, so we have to make some modification.

z(·) is measurable, it is almost a uniform limit of a sequence of countably valued functions. That is, there is a subset C ⊂ B with μC = μB and a sequence of countably valued functions {y_{n}(·)} such that y_{n}(t) → z(t) uniformly on C. There exist a natural number k_{1} satisfying ∥y_{n}(t) − z(t)∥_{X} < \frac{1}{n} whenever n > k_{1}. Denote by y_{n}(t) = ∑ _{i=1}^{∞} e_{in} χ_{C_{in}} ,(μ(C_{in}) > 0). y_{n} is single valued on C_{in}, i.e., y_{n}(t) ≡ e_{in} (t ∈ C_{in}). Let k = max\{k_{0}, k_{1}\}. When n > k, we have 1 − \frac{1}{n} < ∥z_{n}(t) ± z(t)∥_{X} < 1 + \frac{1}{n}, and ∥z_{n}(t)∥_{X} ⩾ δ. Select a point t_{i_{0}} ∈ C_{in}, then define x_{n}(t) = ∑ _{i=1}^{∞} z_{n}(t_{i_{0}} )χ_{C_{in}} . Then {x_{n}(·)} is a sequence of measurable functions since they are countably valued.

For every t ∈ C_{in} (i = 1, 2, …), if n > k, then

∥x_{n}(t) ± z(t)∥_{X} ⩾ ∥x_{n}(t) ± z(t_{i_{0}} )∥_{X} − ∥z(t_{i_{0}} ) − y_{n}(t_{i_{0}} )∥_{X} − ∥y_{n}(t_{i_{0}} ) − z(t)∥_{X}
= ∥z_{n}(t_{i_{0}} ) ± z(t_{i_{0}} )∥_{X} − ∥z(t_{i_{0}} ) − y_{n}(t_{i_{0}} )∥_{X} − ∥y_{n}(t) − z(t)∥_{X}
> 1 − \frac{1}{n} − \frac{1}{n} − \frac{1}{n} = 1 − \frac{3}{n} → 1.
Analogously, ∥x_{n}(t) ± z(t)∥_{X} ⩽ 1 + \frac{1}{n} + \frac{1}{n} + \frac{1}{n} = 1 + \frac{3}{n} → 1. Moreover,

∥x_{n}(t)∥_{X} = ∥z_{n}(t_{i_{0}} )∥_{X} ⩾ δ.

Now we define a sequence of measurable functions on G as follows:

x′_{n}(t) = \{ \begin{matrix}0, & t ∈ G ∖ C \\ ∥x(t)∥_{X} x_{n}(t), & t ∈ C. \end{matrix}

Thus, for n > k,

|∥x′_{n}(t)∥_{X} | = ∥∥x′_{n}(t)∥_{X} ∥_{E}
= ∥∥∥x(t)∥_{X} x_{n}(t)∥_{X} χ_{C} ∥_{E} = ∥∥x(t)∥_{X} ∥x_{n}(t)∥_{X} χ_{C} ∥_{E}
= ∥∥x(t)∥_{X} χ_{C}∥x_{n}(t)∥_{X} ∥_{E} ⩾ δ∥∥x(t)∥_{X} χ_{C} ∥_{E} = δ′ > 0.

On the other hand,

⫴x′_{n}(t) ± x(t)⫴ = ∥∥x′_{n}(t) ± x(t)∥_{X} ∥_{E}

= ∥∥x(t)∥_{X} χ_{G∖C} + ∥x(t) ± ∥x(t)∥_{X} x_{n}(t)∥_{X} χ_{C} ∥_{E}

= ∥∥x(t)∥_{X} χ_{G∖C} + ∥x(t)∥_{X} ∥ \frac{x(t)}{∥x(t)∥_{X}} ± x_{n}(t)∥_{X} χ_{C} ∥_{E}

⩾ ∥∥x(t)∥_{X} χ_{G∖C} + (1 − \frac{3}{n})∥x(t)∥_{X} χ_{C} ∥_{E}

> (1 − \frac{3}{n})∥∥x(t)∥_{X} ∥_{E} = (1 − \frac{3}{n}) ⫴ x(t)⫴ = 1 − \frac{3}{n} → 1.

Similarly,

⫴x′_{n}(t) ± x(t)⫴ ⩽ ∥∥x(t)∥_{X} χ_{G∖C} + (1 + \frac{3}{n})∥x(t)∥_{X} χ_{C} ∥_{E}
⩽ (1 + \frac{3}{n})∥∥x(t)∥_{X} ∥_{E} = (1 + \frac{3}{n}) ⫴ x(t)⫴ = 1 + \frac{3}{n} → 1.

This contradicts the fact that x is a strongly extreme point of B(E(X)). The proof is completed. □

References

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5. L. W. Ren, G. C. Feng, C. X. Wu, Rotundity in Köthe spaces, Adv. Math., Beijing 35(3) (2006), 350–360.

Qiaoguang School (Received 03 11 2010) ZhuHai, Guangdong 519020 P. R. China

hzht2008@163.com

Luneng Vocational Secondary School Linyi, Shandong, 276000 P. R. China

lemonshuang1984@163.com