Extremal Properties of the Set of Vector-Valued Banach Limits

Open Math. 2015; 13: 757–767

Open Mathematics Open Access

Research Article

Francisco Javier García-Pacheco* Extremal properties of the set of vector-valued Banach limits

DOI 10.1515/math-2015-0067 Received June 23, 2015; accepted October 16, 2015.

Abstract: In this manuscript we ﬁnd another class of real Banach spaces which admit vector-valued Banach limits different from the classes found in [6, 7]. We also characterize the separating subsets of ` .X/. For this we ﬁrst 1 need to study when the space of almost convergent sequences is closed in the space of bounded sequences, which turns out to happen only when the underlying space is complete. Finally, a study on the extremal structure of the set of vector-valued Banach limits is conducted when the underlying normed space is a Hilbert space. We also reach the conclusion that the set of vector-valued Banach limits is not a convex component of BCL.` .X/;X/, provided that X 1 is a 1-injective Banach space satisfying that the underlying compact Hausdorff topological space has isolated points.

Keywords: Banach limit, Almost convergence, Group of Isometries, Extremal Structure

MSC: 40J05, 46B15, 46B25

1 Introduction

In his 1932 book Théorie des opérations linéaires (see [1, p. 34]) Banach extended in a natural way the limit function deﬁned on the space of all convergent real sequences to the space of all bounded real sequences.

Definition 1.1 (Banach, [1]). A linear function ' ` R is called a Banach limit exactly when: W 1 ! 1. ' ..x / / 0 if .x / ` and x 0 for all n , n n N n n N n N 2 2 2 1 2 2. ' ..xn/n / ' .xn 1/n if .xn/n ` , and N D C N N 2 1 3. ' 1 12, where 1 denotes the2 constant2 sequence of general term 1. D Banach noticed that a linear function ' ` R is a Banach limit if and only if ' is invariant under the shift W 1 ! operator on ` and lim inf x ' ..x / / lim sup x for all .x / ` . As a consequence, n n n n N n n n n N 1 !1 Ä 2 Ä !1 2 2 1 every Banach limit ' ` R verifies that ' c lim and ' 1. Therefore, Banach limits are norm-1 Hahn- W 1 ! j D k k D Banach extensions of the limit function on c to ` . For a wider perspective on Banach limits we refer the reader 1 to the very interesting contributions given in [2–5]. In [6] the authors fully extend the concept of Banach limit to vector-valued sequences just by noticing that a function ' ` is a Banach limit if and only if ' S , ' is R ` W 1 ! 2 1 invariant under the shift operator on ` , and ' c lim. 1 j D Definition 1.2 (Armario, García-Pacheco, Pérez-Fernández, [6]). Let X be a real normed space. The set of vector- valued Banach limits on X is defined as BL .X/ SN LX . WD X \ If X R, then we simply write BL. The notation used in the definition right above follows. Given a real normed D space X, we have the following:

*Corresponding Author: Francisco Javier García-Pacheco: Department of Mathematics, University of Cadiz, Puerto Real 11510, Spain, E-mail: [email protected]

– CL .` .X/ ;X/ is meant to be the space of all continuous and linear operators from ` .X/ to X when ` .X/ 1 1 1 is endowed with the sup norm.

– NX will denote the set of all continuous and linear operators from ` .X/ to X which are invariant under the 1 shift operator on ` .X/. In other words, 1 ˚ NX T CL .` .X/ ;X/ T ..xn/ / T .xn 1/ n N n N WD 2 1 W 2 D C 2 for all .x / ` .X/ : n n N 2 2 1 g It is clear that NX is a closed vector subspace of CL .` .X/ ;X/ which is also closed when CL .` .X/ ;X/ 1 1 is endowed with the pointwise convergence topology.

– LX will stand for the set of all continuous and linear operators from ` .X/ to X which are extensions of the 1 limit function on c .X/. In other words,

LX T CL .` .X/ ;X/ T c.X/ lim : WD f 2 1 W j D g It is clear that LX is a an affine subset of CL .` .X/ ;X/ (that is, if T;S LX and t R, then tT .1 t/ S 1 2 2 C 2 LX ), which is also closed when CL .` .X/ ;X/ is endowed with the pointwise convergence topology, and 1 every T LX verifies that 2 T T c.X/ lim 1: k k k j k D k k D We will finalize this introduction with the following proposition which will be very useful throughout this manuscript. We recall the reader that if X is a real normed space and x X, then x denotes the constant sequence of 2 general term equal to x. As expected, X denotes the closed subspace of ` .X/ composed of all constant sequences 1 x.

Proposition 1.3 (Armario, García-Pacheco, Pérez-Fernández, [7]). Let X be a real normed space. Then:

1. NX T CL .` .X/ ;X/ bps .X/ ker .T / , where bps .X/ stands for the set of sequences in X with D f 2 1 W Â g bounded partial sums. As a consequence, if T NX , then c0 .X/ ker .T /. 2 Â 2. LX T CL .` .X/ ;X/ c0 .X/ ker .T / and T .x/ x for all x X : D f 2 1 W Â D 2 g 3. NX LX T CL .` .X/ ;X/ bps .X/ ker .T / and T .x/ x for all x \ D f 2 1 W Â D X : 2 g 4. BL .X/ ' SN ' .x/ x for all x X : D f 2 X W D 2 g 5. BL .X/ BNX LX and thus it is convex and closed when CL .` .X/ ;X/ is endowed with the pointwise D \ 1 convergence topology.

2 More on the existence of vector-valued Banach limits

In [6] it was shown that a large class of real Banach spaces admit vector-valued Banach limits. On the contrary, c0 is free of vector-valued Banach limits.

Theorem 2.1 (Armario, García-Pacheco, Pérez-Fernández, [6]). 1. If X a real Banach space 1-complemented in X , then BL .X/ ¿. ¤ 2. BL .c0/ ¿. D In [7] Theorem 2.1 was extended to the following result.

Corollary 2.2 (Armario, García-Pacheco, Pérez-Fernández, [7]). Let X be a real Banach space with a monotone Schauder basis. The following conditions are equivalent: – BL .X/ ¿. ¤ – X is 1-complemented in its bidual.

Another spaces admitting vector-valued Banach limits are described in the following result. Extremal properties of the set of vector-valued Banach limits 759

Theorem 2.3 (Armario, García-Pacheco, Pérez-Fernández, [7]). Let .X / be a sequence of real normed spaces i i N 2 with BL .Xi / ¿ for every i N. Assume that 'i BL .Xi / for every i N. The map ¤ 2 2 2 P P ' ` Xi Xi i N i N W 1 n2 1 ! 2 n 1 x 'i x i i N n N i n N i N 2 2 7! 2 2 P veriﬁes that ' BL Xi . i N 2 2 1 In this section we ﬁnd a different class of real Banach spaces which also admit vector-valued Banach limits.

Lemma 2.4. Let X be a real normed space. If x SX and x SX are so that x .x/ 1, then x .UX .x; "// 2 2 D .1 "; / for all 0 < " 1. C1 Ä

Proof. Because of the convexity of x .UX .x; "// it sufﬁces to show that no element y UX .x; "/ veriﬁes that 2 x .y/ 1 ". Indeed, assume to the contrary that such element y exists. Then D ˇ ˇ ˇ ˇ " ˇx .x/ x .y/ˇ ˇx .x y/ˇ x y < "; D D Ä k k which is a contradiction.

Lemma 2.5. Let X be a real normed space. Then:

1. If x SX , then d .x; bps .X// 1. 2 D 2. bps .X/ X 0 , where bps .X/ denotes the closure of bps .X/ in ` .X/. \ D f g 1

Proof. 1. In the ﬁrst place, we will show that d .x; bps .X// 1. Let x SX such that x .x/ 1. Fix 2 D an arbitrary 0 < " < 1. By applying Lemma 2.4 we have that x .UX .x; "// .1 "; /. Now consider C1 .xn/ bps .X/ such that n N 2 2 .x / x ": n n N k 2 k1 Ä Notice that .x / U .x; "/ therefore x .x / > 1 " for all n . Let M > 0 be such that n n N X n N 2 2 m X xn M Ä n 1 D for all m N. Finally notice that 2 m ! m X X m .1 "/ < x xn xn M Ä Ä n 1 n 1 D D for all m N which is impossible. As a consequence, d .x; bps .X// 1. In order to see that d .x; bps .X// 1 it 2 D only sufﬁces to realize that .x; 0; 0; : : : ; 0; : : : / bps .X/ 2 and .x; 0; 0; : : : ; 0; : : : / x 1: k k1 D 2. It is a direct consequence of 1.

We remind the reader that a Banach space X is said to be 1-injective if it has the following extension property: every bounded linear operator from a subspace of a Banach space into X can be extended to a bounded linear operator on the whole space preserving its norm. We refer the reader to [8, 9] for a wide perspective on injective Banach spaces.

Theorem 2.6. If X is a 1-injective real Banach space, then BL .X/ ¿. ¤ Proof. Consider the following operator: bps .X/ X X ˚ ! (1) .x / x x n n N 2 C 7! 760 F.J. García-Pacheco

Notice that, in virtue of 1. in Lemma 2.5, we have that Á x d x; bps .X/ .x / x n n N k k D Ä k 2 C k1 for all x X and all .xn/ bps .X/. Finally Theorem 1.3 serves to assure that any norm-1 Hahn-Banach 2 n N 2 extension of the operator given2 in Equation (1) is an element of BL .X/.

3 The space of almost convergent sequences

The concept of almost convergence is intimately related to Banach limits. We refer the reader to [10] for a wider perspective on such strong connection. For a complete view on matrix summability methods we refer the reader to the very interesting contributions and expositions given in [11–15]. Here we will focus on the vectorial version of the almost convergence.

Deﬁnition 3.1 (Lorentz, [10]; Boos, [11]). Let X be a real normed space. A sequence .x / is called almost n n N convergent when there exists y X (called the almost limit of .x / ) such that lim 1 2Pp x y n n N p p 1 k 0 n k 2 2 !1 C D C D uniformly in n N. We will denote it by AC limn xn y: 2 !1 D Beware about the following remark on which we will rely later on.

Remark 3.2. Let X be a real normed space. The set of all almost convergent sequences in X is usually denoted by ac .X/. According to [6, 7, 11, 16, 17] we have the following: – ac .X/ is a vector subspace of ` .X/. 1 – c .X/ ac .X/. Â – If X is complete, then ac .X/ is closed in ` .X/. 1 – The almost limit function is deﬁned as

AC lim ac .X/ X W ! (2) .x / AC lim x n n N n 2 7! n !1 and is a norm-1 continuous linear operator such that AC lim c.X/ lim. ˚ « j D – bps .X/ .xn 1 xn/ .xn/ ` .X/ and thus bps .X/ ac0 .X/, where ac0 .X/ denotes the n N n N D C 2 W 2 2 1 Â space of null almost convergent sequences in X, that is, ac0 .X/ ker .AC lim/. D – If .x / ac .X/, then T ..x / / AC lim x for all T BL .X/. n n N n n N n 2 2 2 D n 2 – BL .X/ HB .AC lim/, where HB .AC lim/ denotes!1 the set of norm-1 Hahn-Banach extensions of AC lim to D the whole of ` .X/. 1 In [17, Corollary 2.1] it is proved that a real normed space X is complete if and only if ac .X/ is also complete. The next result shows that a real normed space X is complete if and only if ac .X/ is a closed subspace of ` .X/. We 1 will make use of the following lemma.

Lemma 3.3. Let X be a real normed space. If Y denotes the completion of X, then c .X/ is dense in c .Y /.

Proof. Let .y / c .X/ and ﬁx an arbitrary " > 0. For every n let x X such that n n N N n 2 2 2 2 yn xn "=2. Let us take n0 N such that 1=n0 < "=2 and yp yq < "=2 for all p; q n0. Then k k Ä 2 k k .x1; x2; : : : ; xn0 1; xn0 ; xn0 ; xn0 ;::: / c .X/ and 2 .x ; x ; : : : ; x ; x ; x ; x ;::: / .y / < " 1 2 n0 1 n0 n0 n0 n n N k 2 k1 since " " xn yn xn yn yn yn < " k 0 k Ä k 0 0 k C k 0 k 2 C 2 D for all n n0. Extremal properties of the set of vector-valued Banach limits 761

An interesting question would be to determine whether ac .X/ is dense in ac .Y / under the assumptions of Lemma 3.3.

Theorem 3.4. Let X be a real normed space. The following conditions are equivalent: 1. If Y denotes the completion of X, then ac .Y / ` .X/ ac .X/. \ 1 D 2. ac .X/ is a closed subspace of ` .X/. 1 3. X is complete.

Proof. 1. 2. Immediate if taken into account that ac .Y / is closed in ` .Y / in virtue of [17, Corollary 2.1]. ) 1 2. 3. Assume to the contrary that X is not complete. Consider a non-convergent Cauchy sequence .x / X. n n N ) 2 It is obvious that .xn/n c .Y / ` .X/ ac .Y / ` .X/. In accordance with Lemma 3.3 there exists N 2 \ 1 \ 1 a sequence of elements2 of c .X/ converging to .x / . Since c .X/ ac .X/, we deduce by hypothesis that n n N 2 .xn/ ac .X/, which is impossible since the limit of .xn/ in Y does not belong to X. n N 2 n N 3. 1.2 It is obvious. 2 ) The last part of this section is devoted to show that Theorem 3.4 has a strong inﬂuence of the connection between vector-valued almost convergence and vector-valued Banach limits.

Proposition 3.5. Let X be a real normed space. Define the set [ qc .X/ W ` .X/ T;S BL .X/ T W S W : WD f Â 1 W 2 ) j D j g Then: 1. qc .X/ is a closed vector subspace of ` .X/. 1 2. ac .X/ qc .X/. Â 3. If dim .X/ 1, then ac .X/ qc .X/. D D 4. qc .X/ ` .X/ if and only if BL .X/ is a singleton. D 1 Proof. 1. Obvious from the definition of qc .X/. 2. In accordance with Remark 3.4 we have that BL .X/ HB .AC lim/, which automatically implies that ac .X/ D Â qc .X/. 3. This is a direct consequence of the famous Lorentz Theorem (see [10]), which states that a sequence .x / n n N 2 2 ` is almost convergent to x R if and only if T ..xn/n / x for all T BL. 1 2 R D 2 4. If BL .X/ is a singleton, then it is immediate that qc .X2/ ` .X/. Conversely, assume that qc .X/ ` .X/. D 1 D 1 Let T;S BL .X/ and fix any arbitrary .xn/n ` .X/. By hypothesis, .xn/n qc .X/, therefore 2 N 2 1 N 2 T ..x / / S ..x / /. 2 2 n n N n n N 2 D 2 An interesting question would be to determine whether ac .X/ is dense in qc .X/ under the assumptions of Proposition 3.5. A corollary of Proposition 3.5 provides a solution to this question, provided that there exists a real normed space admitting only one vector-valued Banach limit.

Corollary 3.6. Let X be a real normed space. If BL .X/ is a singleton, then ac .X/ is not dense in qc .X/.

Proof. According to 4. of Proposition 3.5 we have that qc .X/ ` .X/. Finally, by applying [7, Remark 4.1] we D 1 have that ac .X/ is never dense in ` .X/. 1 We would like to make the reader notice that at this stage we are unaware of the existence of a real normed space admitting only one vector-valued Banach limit. The next proposition serves to provide a new expression of the set of vector-valued Banach limits in connection with Remark 3.2.

Proposition 3.7. Let X be a real normed space with BL .X/ ¿. Then: ¤ 1. The QC-limit function, deﬁned as

QC lim qc .X/ X W ! (3) .x / QC lim x T ..x / / n n N n n n N 2 7! n WD 2 !1 762 F.J. García-Pacheco

where T is any element of BL .X/, is a norm-1 continuous linear operator such that QC lim ac.X/ AC lim. j D 2. HB .QC lim/ BL .X/ HB .AC lim/. D D Proof. 1. It is a simple exercise.

2. Attending to Remark 3.4 we already know that BL .X/ HB .AC lim/. On the other hand, QC lim ac.X/ D j D AC lim in virtue of Paragraph 1., therefore HB .QC lim/ HB .AC lim/ BL .X/. Finally, by construction of Â D qc .X/ and QC lim ac.X/, it is clear that BL .X/ HB .QC lim/. j Â On the other hand, the vector-valued version of the Lorentz Theorem was partially accomplished as follows (see [6] and Remark 3.2): If X is any real normed space and .x / ac .X/, then T ..x / / AC lim x for all n n N n n N n 2 2 2 D n T BL .X/. A real normed space X with BL .X/ ¿ is said to verify the vector-valued version!1 of the Lorentz 2 ¤ Theorem when the reverse implication holds, that is, if T ..xn/ / x for all T BL .X/, then .xn/ is n N D 2 n N almost convergent to x. As a corollary of Theorem 3.4 and Proposition2 3.5 we obtain that if a real normed space2 satisﬁes the vector-valued version of the Lorentz Theorem, then it must be complete.

Corollary 3.8. Let X be a real normed space such that BL .X/ ¿. If X veriﬁes the vector-valued version of the ¤ Lorentz Theorem, then X is complete.

Proof. Notice that by hypothesis we have that ac .X/ qc .X/. Now qc .X/ is a closed vector subspace of ` .X/ D 1 according to 1. of Proposition 3.5, which implies that ac .X/ is also closed in ` .X/. Finally, Theorem 3.4 assures 1 that X is complete. Another interesting question related to Corollary 3.8 would be the problem of ﬁnding a non-complete normed space X such that BL .X/ ¿. In relation to this question we have come up with the following result. ¤ Theorem 3.9. Let X be a real normed space and consider a proper subspace Y of X. Then:

1. If T BL .X/, then T ` .Y / BL .Y / if and only if T .` .Y // Y . 2 j 1 2 1 Â 2. If Y is dense in X and S BL .Y /, then no extension T of S to the whole of ` .X/ veriﬁes that T BL .X/. 2 1 2 Proof. 1. It is obvious.

2. Assume on the contrary that T BL .X/ veriﬁes that T ` .Y / S BL .Y /. By Parapraph 1. we obtain that 2 j 1 D 2 T .` .Y // Y . Let .yn/n be a sequence in Y converging to an element x X Y . Under these circumtances 1 Â N 2 n we reach the following contraditction:2

x T ..y / / S ..y / / Y: n n N n n N D 2 D 2 2 The following corollary can be immediately derived from the previous theorem and thus we spare the details of its proof to the reader.

Corollary 3.10. Let Y be a non-complete real normed space and denote by X to its completion. If S BL .Y /, 2 then no extension T of S to the whole of ` .X/ veriﬁes that T BL .X/. 1 2

4 Separating sets

A non-empty subset G of ` is called separating provided that if B1 and B2 are Banach limits such that B1 G 1 j D B2 G , then B1 B2. In [3] the concept of separating set was introduced for the ﬁrst time. This concept can easily j D be transported to the scope of vector valued Banach limits.

Deﬁnition 4.1. Let X be a real normed space. Let G ` .X/ be non-empty. We say that G is separating provided Â 1 that the following condition holds: if T;S BL .X/ are so that T G S G , then T S. 2 j D j D Extremal properties of the set of vector-valued Banach limits 763

This section is aimed at finding sufficient conditions and characterizations of separating sets. The key to finding separating sets rely on a reasonable comprehension of the space of almost convergent sequences.

Proposition 4.2. Let X be a real normed space with BL .X/ ¿. Let G be a non-empty subset ` .X/. Then: ¤ 1 1. If qc .X/ span .G/ is dense in ` .X/, then G is separating. C 1 2. If G qc .X/ and q .X/ ` .X/, then G is not separating. Â ¤ 1

Proof. 1. Let T;S BL .X/ such that T G S G . Notice in the first place that T qc.X/ S qc.X/ by defnition 2 j D j j D j of qc .X/. Now by linearity and continuity we immediately deduce that T S. D 2. It is sufficient to consider any .x / ` .X/ qc .X/. By definition of qc .X/ there must exist T;S n n N 2 2 1 n 2 BL .X/ such that T ..xn/ / S ..xn/ /. However, T G S G because G qc .X/, therefore G is not n N ¤ n N j D j Â separating. 2 2

We would like to make the reader recall that, given any real normed space X with BL .X/ ¿, the fact that ¤ qc .X/ ` .X/ is equivalent to the fact that BL .X/ is a singleton (see 4. of Proposition 3.5). The following two D 1 theorems characterize the separating sets.

Theorem 4.3. Let X be a real normed space with BL .X/ ¿. The following conditions are equivalent: ¤ 1. BL .X/ is a singleton. 2. Every non-empty subset of ` .X/ is separating. 1 3. Every non-empty subset of qc .X/ is separating. 4. Every non-empty subset of ac .X/ is separating. 5. There exists a non-empty subset of ac .X/ which is separating 6. There exists a non-empty subset of qc .X/ which is separating.

Proof. The reader may realize that all the implications 1. 2. 3. 4. 5. 6. are immediate. Thus we will ) ) ) ) ) show the implication 6. 1. Assume that G qc .X/ is separating. Let T;S BL .X/. By deﬁnition of qc .X/ ) Â 2 we have that T G S G and hence T S by assumption on G. j D j D The previous result takes care of the case of only one vector-valued Banach limit. The following one takes care of the case of more than one vector-valued Banach limit.

Theorem 4.4. Let X be a real normed space with BL .X/ ¿. For every T;S BL .X/ consider the set ¤ 2 V .x / ` .X/ T ..x / / S ..x / / : T;S n n N n n N n n N WD f 2 2 1 W 2 D 2 g T 1. qc .X/ VT;S T;S BL .X/ . D f W 2 g 2. If BL .X/ is not a singleton, then a non-empty subset G of ` .X/ is separating if and only if G 1 \ .` .X/ VT;S / ¿ for all T S BL .X/. 1 n ¤ ¤ 2 Proof. 1. Immediate.

2. Assume that G is separating. Let T S BL .X/. By assumption there must exist .xn/ G such that ¤ 2 n N 2 T ..x / / S ..x / /, which means that .x / G .` .X/ V /. Conversely,2 assume that G n n N n n N n n N T;S 2 ¤ 2 2 2 \ 1 n \ .` .X/ VT;S / ¿ for all T S BL .X/. Let T;S BL .X/ such that T G S G . By assumption on G 1 n ¤ ¤ 2 2 j D j we have that T S. D As an immediate corollary we obtain a sharp characterization of non-separating sets.

Corollary 4.5. Let X be a real normed space with BL .X/ ¿. If BL .X/ is not a singleton, then a non-empty ¤ subset G of ` .X/ is not separating if and only if G VT;S for some T S BL .X/. 1 Â ¤ 2 We would like to ﬁnish this section by posing the question of ﬁnding, if they exist, minimal separating sets. 764 F.J. García-Pacheco

5 Extremal structure of the set of vector-valued Banach limits

The extremal structure of the set of Banach limits has been studied in [7] where the following result was proved. We remind the reader that a convex subset F of a convex set C is said to be a face of C provided that F veriﬁes the extremal condition: if c; d C , t .0; 1/, and tc .1 t/ d F , then c; d F . We also remind the reader that 2 2 C 2 2 if X is a real normed space, then GX denotes the group of surjective linear isometries on X.

Theorem 5.1 (Armario, García-Pacheco, Pérez-Fernández, [7]). Let X be a real normed space such that BL .X/ ¿. ¤ 1. If X has the Bade property, that is, BX co .ext .BX //, then BL .X/ is a face of BN . D X 2. If there exists T GX IX with a ﬁxed point x SX , then BL .X/ is not a convex component of SN . 2 n f g 2 X We remind the reader that a convex component is a maximal convex subset and refer the reader to [18] for a deep perspective on applications of convex components. Taking into account that any real Hilbert space of dimension greater than or equal to 3 enjoys the Bade Property (since it is strictly convex) and has a surjective linear isometry other than the identity with non-zero ﬁxed points, we have the following corollary.

Corollary 5.2 (Armario, García-Pacheco, Pérez-Fernández, [7]). If X is a real Hilbert space with dim .X/ 3, then BL .X/ is a non-maximal face of BNX .

Bearing in mind the Banach-Stone Theorem and 2. of Theorem 5.1, we ﬁnd more spaces X for which BL .X/ is not a convex component of SNX .

Theorem 5.3. Let K be a compact Hausdorff topological space. Assume that K has an isolated point k0 satisfying that there exists a homeomorphism ' K k0 K k0 different from the identity. Then: W n f g ! n f g 1. The map C .K; R/ C .K; R/ ! K R ! ( f f .k0/ if k k0 7! k D 7! ' .f .k// if k k0 ¤

is an element of GC.K;R/ IC.K;R/ having k0 as a ﬁxed point. n f g f g 2. BL .C .K; // is not a convex component of SN . R C.K;R/

We recall the reader that a real Banach space X is 1-injective if and only if X is isometrically isomorphic to a C .K; R/ for some extremally disconnected compact Hausdorff topological space K (see [19, Page 123]). This section is aimed at showing that if X C .K; R/ for K a extremally disconnected compact Hausdorff topological space with isolated WD points, then BL .X/ is not a convex component of BCL.` .X/;X/: 1 Lemma 5.4. Let X be a real normed space. Assume that M and N are closed subspaces of X which are complemented in X. Then: 1. bps .M / and bps .N / are closed subspaces of bps .X/ which are also complemented in bps .X/. 2. If, in addition, X M N , then bps .X/ bps .N / bps .M / when bps .X/ is endowed with the D ˚1 D ˚1 following norm: k X .x / sup x : n n N n k 2 k WD k N n 1 2 D Proof. 1. Let P X M be a continuous linear projection. It is not hard to check that W ! bps .X/ bps .M / ! .x / .P .x // n n N n n N 2 7! 2 is well deﬁned and a continous linear projection of norm P whose kernel is bps .N /. k k Extremal properties of the set of vector-valued Banach limits 765

2. Simply notice that if .x / bps .X/ and x m n with m M and n N for all i , then i i N i i i i i N 2 2 D C 2 2 2 k X .x / sup x i i N i k 2 k D k N i 1 2 D k k X X sup mi ni D C k N i 1 i 1 2 D D ( k k )! X X sup max mi ; ni D k N i 1 i 1 2 D D ( k k ) X X max sup mi ; sup ni D k N i 1 k N i 1 2 D 2 D max .m / ; .n / : i i N i i N D fk 2 k k 2 kg The following theorem is crucial for our purposes. We recall the reader that, given a real normed space X and a vector x SX , we say that x is an L -summand vector of X provided that there exists a closed maximal subspace 2 1 of X such that X Rx M . The reader may observe that Lemma 5.4 is implicitly used in the statement of the D ˚1 following theorem.

Theorem 5.5. Let X be a real normed space. Let x0 SX be an L -summand vector of X and write X 2 1 D Rx M . Consider the continous linear operators ˚1

T1 bps .X/ X X W ˚ ! . x m / x T .. x m / x/ x x n 0 n n N 1 n 0 n n N 1 0 C 2 C 7! C 2 C WD C and S1 bps .X/ X X W ˚ ! . x m / x S .. x m / x/ x x: n 0 n n N 1 n 0 n n N 1 0 C 2 C 7! C 2 C WD C Then: T1 S1 1. If B B .X/, then B bps.X/ X C2 . 2 j ˚ D 1 2. ˛T1 .1 ˛/ S1 1 for all ˛ ; 1 . k C k D 2 2 3. 2 S1 3. Ä k k Ä 4. If T;S ` .X/ X are linear and continuous extensions of T1 and S1, respectively, then: W 1 ! (a) T;S NX LX . T S… [ (b) NX LX . C2 2 \ Proof. 1. Immediate.

T1 S1 2. It is fairly obvious that C 1. We will show now that T1 1, for which it sufﬁces to prove that 2 D k k D T1 1. Let x X and write x x0 m with R and m M . In accordance with 1. of Lemma 2.5 we k k Ä 2 D C 2 2 have that m d .m; bps .M //. Observe the following: k k D

1x0 x .1 / x0 m k C k D k C C k max 1 ; m D fj C j k kg sup n ; m mn Ä n N fj C j k C kg 2 . x m / x : n 0 n n N D k C 2 C k1

Notice that .mn/ bps .M / in accordance with Lemma 5.4. This shows that T1 1. By using the triangular n N 2 k k D inequality we obtain2 that

T1 S1 ˛T1 .1 ˛/ S1 .2˛ 1/ T1 .2 2˛/ C 1 k C k D C 2 Ä

1 1 for all ˛ ; 1 , which in fact means that ˛T1 .1 ˛/ S1 1 for all ˛ ; 1 . 2 2 k C k D 2 2 766 F.J. García-Pacheco

3. By using again 1. of Lemma 2.5 to accomplish that x . x m / x , we have that n 0 n n N k k Ä k C 2 C k1

1x0 x 1x0 x 2 x k C k Ä k k C k k . x m / x 2 . x m / x n 0 n n N n 0 n n N D k C 2 C k1 C k C 2 C k1 3 . x m / x : n 0 n n N D k C 2 C k1

This shows that S1 3. In order to see that S1 2 it only sufﬁces to take into consideration that k k Ä k k

S1 .. x0; 0; 0; : : : ; 0; : : : / x0/ 2x0: C D 4. We will divide this paragraph in two parts:

(a) We will show that T NX LX . In a similar way it can be proved that S NX LX . Simply notice that … [ … [ T .x0; 0; 0 : : : ; 0; : : : / x0 0, therefore D ¤ – T LX because .x0; 0; 0 : : : ; 0; : : : / is a convergent sequence to 0, and … – T NX because .x0; 0; 0 : : : ; 0; : : : / is a sequence with bounded partial sums. … (b) Let .x / bps .X/ and write x x m where and m M for all n . We have that n n N n n 0 n n R n N 2 2 D C 2 2 2 T S 1 ..x / / . x x / 0; C n n N 1 0 1 0 2 2 D 2 D

T S T S Á therefore NX in virtue of 1. of Proposition 1.3. Notice that c0 .X/ ker according to 1. of C2 2 Â C2 T S T S Á Proposition 1.3, thus in order to show that LX it only sufﬁces to prove that .x/ x for all C2 2 C2 D x X, which is immediate by construction of both T1 and S1. 2

The following establishes a sufﬁcient condition for a space of continuous functions to have an L -summand vector. 1

Lemma 5.6. Let K be a compact Hausdorff topological space. If K has an isolated point, then C .K; R/ has an L -summand vector. 1 Proof. Let k0 K be an isolated point. Notice that k0 SC.K;R/, ık0 SC.K; / , ık0 k0 1, and 2 f g 2 2 R f g D C .K; R/ R k0 ker ık0 ; D f g ˚ where k0 denotes the characteristic function of k0 and ık0 is the evaluation functional. Let f C .K; R/ and f g f g 2 write f f .k0/ k0 f f .k0/ k0 : D f g C f g Then we have that

f max f .K/ k k1 D j j max f .k0/ ; max f .K k0 / D fj j j j n f g g ˚ « max f .k0/ k0 ; f f .k0/ k0 : D f g 1 f g 1

This shows that k0 is an L -summand vector of C .K; R/. f g 1 The following ﬁnal corollary concludes the section and the paper.

Corollary 5.7. Let K be a extremally disconnected compact Hausdorff topological space with an isolated point k0. If X C .K; R/, then BL .X/ is not a convex component of BCL.` .X/;X/: WD 1

Proof. By Lemma 5.6 we have that x0 k0 is an L -summand vector of X. In accordance with 1. of Theorem WD f g 1 5.5 we have that the continous linear operators

T1 bps .X/ X X W ˚ ! . x m / x T .. x m / x/ x x n 0 n n N 1 n 0 n n N 1 0 C 2 C 7! C 2 C WD C Extremal properties of the set of vector-valued Banach limits 767 and S1 bps .X/ X X W ˚ ! . x m / x S .. x m / x/ x x n 0 n n N 1 n 0 n n N 1 0 C 2 C 7! C 2 C WD C

T1 S1 satisfy that T1 C 1 and 2 S1 3. In virtue of [19, Page 123] we have that X is 1-injective, k k D 2 D Ä k k Ä therefore we can ﬁnd a norm-preserving Hahn-Banach extension T ` .X/ X of T1. Attending to 2. of W 1 ! Theorem 5.5, we deduce that T SCL.` .X/;X/ but T BL .X/ (in fact, T NX LX ). Now we will prove that 2 1 … … [ BL .X/ is not a convex component of BCL.` .X/;X/: For this it is sufﬁcient to show that if B BL .X/, then the 1 2 segment joining T and B lies entirely in SCL.` .X/;X/. Let ˛ Œ0; 1. We know that ˛T .1 ˛/ B 1 since 1 2 k C k Ä T;B SCL.` .X/;X/. According to 1. and 2. of Theorem 5.5 simply notice that 2 1

T1 S1 .˛T .1 ˛/ B/ bps.X/ X ˛T1 .1 ˛/ C 1; C j ˚ D C 2 D which automatically implies that ˛T .1 ˛/ B 1. k C k D

References

[1] Banach, S., Théorie des opérations linéaires, Chelsea Publishing company, New York, 1978 [2] Ahmad, Z.U., Mursaleen, M., An application of Banach limits, Proc. Amer. Math. Soc., 1988, 103, 244-246 [3] Semenov, E., Sukochev, F., Extreme points of the set of Banach limits, Positivity, 2013, 17, 163-170 [4] Semenov, E., Sukochev, F., Invariant Banach limits and applications, J. Funct. Anal., 2010, 259, 1517-1541 [5] Semenov, E., Sukochev, F., Usachev, A., Structural properties of the set of Banach limits, Dokl. Math., 2011, 84, 802-803 [6] Armario, R., García-Pacheco, F.J., Pérez-Fernández, F.J., On Vector-Valued Banach Limits, Funct. Anal. Appl., 2013, 47, 315-318 [7] Armario, R. García-Pacheco, F.J., Pérez-Fernández, F.J., Fundamental Aspects of Vector-Valued Banach Limits, Izv. Math., 2016, 80, (in press) [8] Rosenthal, H., On injective Banach spaces and the spaces C .S/, Bull. Amer. Math. Soc., 1969, 75, 824-828 [9] Wolfe, J., Injective Banach spaces of continuous functions, Trans. Amer. Math. Soc., 1978, 235, 115-139 [10] Lorentz, G., A contribution to the theory of divergent sequences, Acta Math., 1948, 80, 167-190 [11] Boos, J., Classical and Modern Methods in summability, Oxford University Press, 2000 [12] Mursaleen, M., On some new invariant matrix methods of summability, Quart. Jour. Math. Oxford, 1983, 34, 77-86 [13] Mursaleen, M., On A-invariant mean and A-almost convergence, Analysis Mathematica, 2011, 37, 173-180 [14] Mursaleen, M., Applied Summability Methods, Springer Briefs, Heidelberg New York Dordrecht London, 2014 [15] Raimi, R.A., Invariant means and invariant matrix methods of summability, Duke Math. J., 1963, 30, 81-94 [16] Aizpuru, A., Armario, R., García-Pacheco, F.J., Pérez-Fernández, F.J., Banach limits and uniform almost summability, J. Math. Anal. Appl., 2011, 379, 82-90 [17] Aizpuru, A., Armario, R., García-Pacheco, F.J., Pérez-Fernández, F.J., Vector-Valued Almost Convergence and Classical Properties in Normed Spaces, Proc. Indian Acad. Sci. Math., 2014, 124, 93-108 [18] García-Pacheco, F.J., Convex components and multi-slices in topological vector spaces, Ann. Funct. Anal., 2015, 6, 73-86 [19] Day, M.M., Normed linear spaces, 3rd edition, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 21, Springer-Verlag, New York-Heidelberg, 1973