Lifting Convex Approximation Properties and Cyclic Operators with Vector Lattices رﻓﻊ ﺧﺼﺎﺋﺺ اﻟﺘﻘﺮﯾﺐ اﻟﻤﺤﺪب واﻟﻤﺆﺛﺮات اﻟﺪورﯾﺔ ﻣﻊ ﺷﺒﻜﺎت اﻟﻤﺘﺠﮫ

University of Sudan for Science and Technology College of graduate Studies

A thesis Submitted in Partial Fulfillment of the Requirement of the Master Degree in Mathematics

By: Marwa Alyas Ahmed Supervisor: Prof. Shawgy Hussein Abdalla

November 2016

Dedication

I dedicate my dissertation work to my family, a special feeling of gratitude to my loving parents, my sisters and brothers. I also dedicate this dissertation to my friends and colleagues.

ACKNOWLEDGEMENTS

First of all I thank Allah.. I wish to thank my supervisor Prof. Shawgi huSSien who was more than generous with his expertise and precious time. A special thanks to him for his countless hours of reflecting, reading, encouraging, and most of all patience throughout the entire process. I would like to acknowledge and thank Sudan University for Sciences & Technology for allowing me to conduct my research and providing any assistance requested. Special thanks goes to the all members of the university for their continued support.

Abstract

We demonstrate that rather weak forms of the extendable local reflexivity and of the principle of local reflexivity are needed for the lifting of bounded convex approximation properties from Banach spaces to their dual spaces. We show that certain adjoint multiplication operators are convex- cyclic and show that some are convex- cyclic but no convex polynomial of the operator is hypercyclic. Also some adjoint multiplication operators are convex- cyclic but not 1-weakly hypercyclic. We deal with two weaker forms of injectivity which turn out to have a rich structure behind: separable injectivity and universal separable injectivity. We show several structural and stability properties of these classes of Banach spaces. We provide natural examples of separably injective spaces, including ℒ ultraproducts built over countably incomplete ultrafilters, in spite of the fact that these ultraproducts are never injective. We show that the Fremlin tensor product is not square mean complete when the two spaces are uncountable metrizable compact spaces.

III

اﻟﺨـــﻼﺻـــﺔ

ﺷﺮﺣﻨﺎ أن اﻟﺼﯿﻎ اﻟﻀﻌﯿﻔﺔ اﻷﺧﺮى ﻟﻺﻧﻌﻜﺎﺳﯿﺔ اﻟﻤﻮﺿﻌﯿﺔ اﻟﻘﺎﺑﻠﺔ ﻟﻠﺘﻤﺪﯾﺪ وﻟﻤﺒﺪأ اﻹﻧﻌﻜﺎﺳﯿﺔ اﻟﻤﻮﺿﻌﻲ ھﻲ ﻣﻄﻠﻮﺑﺔ ﻷﺟﻞ اﻟﺮﻓﻊ ﻟﺨﺼﺎﺋﺺ اﻟﺘﻘﺮﯾﺐ اﻟﻤﺤﺪب اﻟﻤﺤﺪود ﻣﻦ ﻓﻀﺎءات ﺑﺎﻧﺎخ إﻟﻰ ﻓﻀﺎءاﺗﮭﺎ اﻟﺜﻨﺎﺋﯿﺔ. أوﺿﺤﻨﺎ أن ﻣﺆﺛﺮات ﺿﺮب اﻟﻤﺮاﻓﻖ اﻟﻤﻌﯿﻦ ھﻮ ﻣﺤﺪب - دوري وأوﺿﺤﻨﺎ أن ﺑﻌﻀﮭﺎ ﻣﺤﺪب- دوري وﻟﻜﻦ ﻟﯿﺲ ﻛﺜﯿﺮة ﺣﺪود ﻣﺤﺪﺑﺔ ﻟﻠﻤﺆﺛﺮ ھﻲ دورﯾﺔ ﻓﻮﻗﯿﺔ. أﯾﻀﺎ ً ﺑﻌﺾ ﻣﺆﺛﺮات اﻟﻀﺮب ھﻲ ﻣﺤﺪﺑﺔ - دورﯾﺔ وﻟﻜﻨﮭﺎ ﻟﯿﺴﺖ دورﯾﺔ ﻓﻮﻗﯿﺔ ﺿﻌﯿﻔﺔ- 1. ﺗﻔﺎﻋﻠﻨﺎ ﻣﻊ ﺻﯿﻐﺘﯿﻦ ﺿﻌﯿﻔﺘﯿﻦ ﺷﺎﻣﻠﺘﯿﻦ واﻟﺘﻲ ﺗﺤﻮل ﻟﺘﻨﺘﺞ ا ﺗﺸﯿﯿﺪً ﻏﻨﯿﺎ ً ﻓﻲ اﻟﺨﻠﻒ: ﺷﻤﻮﻟﯿﺔ ﻣﻨﻔﺼﻠﺔ وﺷﻤﻮﻟﯿﺔ ﻣﻨﻔﺼﻠﺔ ﻋﺎﻟﻤﯿﺔ. أوﺿﺤﻨﺎ إﻧﺸﺎﺋﯿﺔ ﻣﺘﻌﺪدة وﺧﺼﺎﺋﺺ اﺳﺘﻘﺮار ﻟﮭﺬه اﻟﻌﺎﺋﻼت ﻟﻔﻀﺎءات ﺑﺎﻧﺎخ.

أﺷﺘﺮطﻨﺎ أﻣﺜﻠﺔ طﺒﯿﻌﯿﺔ ﻟﻠﻔﻀﺎءات اﻟﺸﺎﻣﻠﺔ اﻟﻤﻨﻔﺼﻠﺔ واﻟﻤﺘﻀﻤﻨﺔ اﻟﻀﺮب اﻟﻔﺎﺋﻖ ℓ اﻟﻤﺒﻨﻲ ﻓﻮق اﻟﻤﺮﺷﺤﺎت اﻟﻔﺎﺋﻘﺔ ﻏﯿﺮ اﻟﺘﺎﻣﺔ اﻟﻘﺎﺑﻠﺔ ﻟﻠﻌﺪ وﺑﺎﻟﺮﻏﻢ ﻣﻦ اﻟﺤﻘﯿﻘﺔ ان ھﺬا اﻟﻀﺮب اﻟﻔﺎﺋﻖ ﻟﯿﺲ ﺷﺎﻣﻼ ً. أوﺿﺤﻨﺎ أن ﺿﺮب ﺗﻨﺴﻮر ﻓﺰﯾﻤﻠﯿﻦ ﻟﯿﺲ ﺗﺎﻣﺎ ً ﻣﺘﻮﺳﻄﺎ ً ﻣﺮﺑﻌﺎ ً ﻋﻨﺪﻣﺎ اﻟﻔﻀﺎﺋﯿﻦ ھﻤﺎ ﻓﻀﺎﺋﯿﻦ ﻣﺘﺮاﺻﯿﻦ ﻗﺎﺑﻠﯿﻦ ﻟﻠﻤﺘﺮﯾﺔ ﻏﯿﺮ اﻟﻘﺎﺑﻠﺔ ﻟﻠﻌﺪ .

TABLE OF CONTENTS

Subject Page Dedication i Acknowledgements ii Abstract iii Abstract (Arabic) iv Table of Contents v List of Symbols vi Chapter 1: Banach Spaces with their Dual Spaces and Related Local 1 Reflexivity Section (1.1): Lifting of the Convex Bounded Approximation Property 1 Section (1.2): Extendable Local Reflexivity Implied by Convex Approxi- 5 mation Properties Chapter 2: Convex Cyclic Operators 10 Section (2.1): The Hahn-Banach Characterization for Convex-Cyclicity 10 Section (2.2): -hypercyclic operators versus convex-cyclic operators with 13 diagonal operators and adjoint multiplication operators Chapter 3: Separably Injective Banach Spaces 21 Section (3.1): Basic properties of separably injective spaces with examples 21 Section (3.2): Two characterizations of universally and 1-separably 33 injective spaces Chapter 4: Complex Vector Lattices 41 Section (4.1): Vector Lattices Complexifications 41 Section (4.2): The Archimedean Vector Lattice Tensor Product 44 References 56

LIST OF SYMBOLS

Symbol Page Sup: Supremum 2 ELR: Extendable Local Reflexivity 2 PLR: Principle Local Reflexivity 2 Inf: infimum 10 Re: Real 10 Orb: orbit 11 : Lebesgue Space of Sequences 13 ⊗: tensor product 15 Im: imaginary 16 Diag: diagonal 16 : Hilbert Space of Sequences 17 H: Hilbert Space 18 : Bergman Spaces 20

∞: Essential Lebesgue Space 21 ext: extension 21 BAP: Bounded Approximation Property 29 ⊕: Orthogonal Sum 29 dim: dimension 31 ker: Kernel 33 max: Maximum 33 van: Vanage 33

: Lebesgue on the Real Line 36 dom: Domain 38 det: Determinal 53

Chapter 1 Banach Spaces with their Spaces and the Related Local Reflexivity We provide a unified approach to the lifting of various bounded approximation properties, including, besides the classical ones, the approximation property of pairs properties of Banach spaces, and the positive approximation property of Banach Lattices. Section (1.1): Lifting of the Convex Bounded Approximation Property Approximation property is a locally convex topological vector spaces is said to have the approximation property if the identity on map can be approximated, uniformly on the compact set, by continuous linear map of finite rank [5]). Let X be a Banach space and let A be an arbitrary subset of ℒ(X). The space X has the -approximation property if for every compact subset of and every > 0, there exists ∈ such that ‖ − ‖ ≤ for all ∈ . Let 1 ≤ < ∞ . The space has the -bounded - approximation property if can be chosen with ‖‖ ≤ (meaning that has the ∩ ℒ()-approximation property). In the case when the set is convex and contains 0, we speak about convex approximation properties. The positive approximation property is precisely the - approximation property where is the cone ℱ() of positive finite-rank operators. The approximation property for pairs is also a convex approximation property. By standard arguments (e.g., that the topology of uniform convergence on compact sets coincides with the strong operator topology on bounded subsets of operators), has the -bounded -approximation property if and only if there exists a net () ⊂ with ‖‖ ≤ for all , and → pointwise, i.e., in the strong operator topology. We say that has the -bounded duality -approximation property if the net () can ∗ ∗ be chosen so that also → ∗ pointwise. The dual space of is said to have the -approximation property with conjugate operators if ∗ has the ∗-approximation property. In the case of convex approximation properties, the following simple result is useful.

Proposition (1.1.1) [1] Let be a Banach space and let be a convex subset of ℒ() containing 0. Let 1 ≤ < ∞ . Then the following properties are equivalent. (a) has the -bounded duality -approximation property. (b) ∗ has the -bounded -approximation property with conjugate operators.

(c) There exists a net () ⊂ such that lim sup‖‖ ≤ and ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗∗ ( ) → ( ) ∀ ∈ , ∀ ∈ . Proof: The implications () ⇒ () ⇒ () are obvious. For the implication () ⇒ (), we apply the (rather straightforward) fact that the -bounded duality -approximation property is equivalent to the ( + )- bounded duality -approximation property for all > 0. To show the latter property, we use that the weak and strong operator topologies on a space of the bounded linear operators yield the same dual space [1]. Hence, having a net converging in the weak operator topology on ℒ() (or on ℒ(∗) as in ()) means that, by passing to convex combinations, one may always assume that the net converges in the strong operator topology on ℒ() (or on ℒ(∗)) to the same element. We introduce the following general forms of the ELR and the PLR. In Theorem (1.1.5) we shall see that these rather weak forms of the ELR and the PLR are sufficient for the lifting of different bounded approximation properties from Banach spaces to their dual spaces. Definition (1.1.2) [1] Let be a Banach space and let be a subset of ℒ(∗∗). Let 1 ≤ < ∞. We say that is -extendably locally reflexive of type if for all finite-dimensional subspaces ⊂ ∗∗ and ⊂ ∗, and for all > 0, there exists ∈ such that () ⊂ , ‖‖ ≤ + , and ∗ ∗∗ ∗∗ ∗ ∗∗ ∗ | ( ) − ( )| ≤ ∀ ∈ , ∀ ∈ . The -ELR of a Banach space clearly implies the -ELR of type ℒ(∗∗). More examples will be presented. Definition (1.1.3) [1] Let be a Banach space, let and be subsets of ℒ() and ℒ(∗∗), respectively. We say that the principle of local reflexivity of type → holds in if for all ∈ , for all finite- dimensional subspaces ⊂ ∗∗ and ⊂ ∗ , and for all > 0, there exists ∈ such that ‖‖ ≤ ‖‖ + and

∗∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ |( )( ) − ( )| ≤ ∀ ∈ , ∀ ∈ . The PLR of type → means that the operators on ∗∗ of “type ” are “locally” of “type ” on . Examples (1.1.4) [1] The following assertions are true. (i) By the PLR, in every Banach space , the PLR of type ℱ(∗∗) → ℱ() holds. (ii) Let be a Banach space and let be a closed subspace of . By the PLR respecting subspaces, the PLR of type { ∈ ℱ(∗∗): () ⊂ } → { ∈ ℱ(): () ⊂ } holds in . ∗∗ (iii) In every Banach lattice , the PLR of type ℱ( ) → ℱ() holds. (iv) Let be a subset of ℒ(). Trivially, in every Banach space , the PLR of type ∗∗ → holds. Theorem (1.1.5) [1] Let be a Banach space. Let be a convex subset of ℒ() containing 0. Let and be subsets of ℒ(∗∗) such that ∗∗ ∘ ⊂ . Let 1 ≤ , < ∞. Assume that the principle of local reflexivity of type → holds in . If is -extendably locally reflexive of type and has the -bounded -approximation property, then has the - bounded duality -approximation property.

Proof: By Proposition (1.1.1) (c), it suffices to construct a net () ⊂ such that lim sup‖‖ ≤ and ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗∗ ( ) → ( ) ∀ ∈ , ∀ ∈ . Consider the set of all = (, , ), where ⊂ ∗∗ and ⊂ ∗ are finite-dimensional subspaces and > 0, directed in the natural way.

Since is -ELR of type , for every , there exists an operator ∈ such that () ⊂ , ‖‖ ≤ + , and ∗ ∗∗ ∗∗ ∗ ∗∗ ∗ | ( ) − ( )| ≤ ∀ ∈ , ∀ ∈ . The set () ⊂ is compact because is compact. Since has the -bounded -approximation property, there exists ∈ with ‖‖ ≤ such that ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ ‖ − ‖ = ‖ − ‖ ≤ ∀ ∈ . ∗∗ ∗∗ We have ∈ ∘ ⊂ . By the PLR of type → , there exists ∈ with ∗∗ ‖‖ ≤ ‖ ‖ + ≤ ( + ) + , implying that

lim sup‖‖ ≤ ,

3 and ∗∗ ∗∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ |( )( ) − ( )| ≤ ∀ ∈ , ∀ ∈ . ∗∗ ∗ For arbitrary ∈ and ∈ , we have ∗∗ ∗ ∗ ∗∗ ∗ | ( ) − ( )| ∗∗ ∗ ∗ ∗∗ ∗∗ ∗ ≤ | ( ) − ( ) ( )| ∗∗ ∗∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗∗ ∗ + |( ) ( ) − ( )( )| + | ( ) − ( )| ≤ 3. This implies that ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗∗ lim ( ) = ( ) ∀ ∈ , ∀ ∈ . ∗ ∗∗ Indeed, let ∈ ∗ and ∈ ∗∗ be given. For > 0 , take = ∗∗ ∗ ∗∗ ∗ (span { }, span{ }, /3). If = (, , ) ≥ , then ∈ , ∈ , ≤ /3, and we have ∗∗ ∗ ∗ ∗∗ ∗ | ( ) − ( )| ≤ 3 ≤ , as needed. From Theorem (1.1.5) and Example (1.1.4) (iv), we have the following immediate corollary that will be applied in lifting results. Corollary (1.1.6) [1] Let be a Banach space. Let be a convex subset of ℒ() containing 0 and let be a subset of ℒ(∗∗) such that ∗∗ ∘ ⊂ ∗∗. Let 1 ≤ , < ∞. If is -extendably locally reflexive of type Cand has the -bounded -approximation property, then has the -bounded duality -approximation property.

Section (1.2): Extendable Local Reflexivity Implied by Convex Approximation Properties The lifting Theorem (1.2.1) by Johnson and Oikhberg has a strong converse, due to Rosenthal. Theorem (1.2.1) [1] (Rosenthal). Let be a Banach space. Let 1 ≤ < ∞ . If ∗ has the -bounded approximation property, then is - extendably locally reflexive. Recall that, by an important result, the assumption “∗ has the - bounded approximation property” is equivalent to “∗ has the -bounded approximation property with conjugate operators”. We shall see that Rosenthal’s Theorem (1.2.1) can be extended as follows, providing a general converse to our main Theorem (1.1.5). Proposition (1.2.2) [1] Let be a Banach space. Let Abe a subset of () . Let 1 ≤ < ∞ . If ∗ has the -bounded -approximation property with conjugate operators, then is -extendably locally reflexive of type ∗∗. Proof: Let ⊂ ∗ be a finite-dimensional subspace and let > 0. Since ∗ ∗ is compact and has the -bounded { : ∈ } -approximation property, there exists ∈ with ‖‖ ≤ such that ∗ ∗ ∗ ∗ ‖ − ‖ ≤ ∀ ∈ . Then ∗∗ ∈ ∗∗ and ‖∗∗‖ ≤ . Since ∈ () , we have that ∗∗ ∗∗ ∗∗ ∗ ( ) ⊂ [1]. For arbitrary ∈ ∗∗ and ∈ , |∗(∗∗∗∗) − ∗∗(∗)| = |∗∗(∗∗) − ∗∗(∗)| ≤ ‖∗∗‖‖∗∗ − ∗‖ ≤ . Hence, for any finite-dimensional subspace of ∗∗, the conditions of the -ELR of type ∗∗ for are satisfied. In the case of Banach lattices, we have the following version of Rosenthal’s Theorem (1.2.1). Theorem (1.2.3) [1] Let be a Banach lattice. Let 1 ≤ < ∞. If the dual lattice ∗ has the -bounded positive approximation property, then is positively -extendably locally reflexive. Proof: Since ∗ has the -bounded positive approximation property, it has the -bounded positive approximation property with conjugate ∗∗ ∗∗ operators [1]. Let = ℱ() . Then ⊂ () and ⊂ ℱ( ) ⊂ ∗∗ ℒ( ). The claim is immediate from Proposition (1.2.2). The following result is immediate from Proposition (1.2.2) and Corollary (1.1.6). It shows that the lifting of convex approximation

5 properties from a Banach space to its dual space is possible whenever the dual space already enjoys a weaker approximation property. Theorem (1.2.4) [1] Let be a Banach space. Let be a convex subset of ℒ() containing 0 and let be a subset of () such that ∘ ⊂ . Let 1 ≤ , < ∞ . If ∗ has the -bounded -approximation property with conjugate operators and has the -bounded -approximation property, then has the -bounded duality -approximation property. If, in Theorem (1.2.4), ⊂ () and ∗ or ∗∗ has the Radon– Nikodym property, then has the metric duality -approximation property. Let us have general application of Theorem (1.2.4) to (positive) approximation properties of pairs. Corollary (1.2.5) [1] Let be a Banach space and a closed subspace of . Let be an operator ideal. Denote = { ∈ (): () ⊂ } and = { ∈ (): () ⊂ }. Let 1 ≤ , < ∞. Then the assertion of Theorem (1.2.4) holds. In the special case when is a Banach lattice, and may be replaced by and . The classical cases when Corollary (1.2.5) applies are = ℱ() and = (). For instance, it follows that the dual lattice ∗ has the bounded (metric if ∗ has the Radon–Nikodym property) positive approximation property whenever has the bounded positive approximation property and ∗ has the bounded positive weakly compact approximation property with conjugate operators. If one adds in the definition of the -ELR the requirement that the operator ∈ ℒ(∗∗) also satisfies ∗(∗) ⊂ ∗ , then one obtains the notion of the strong -extendable local reflexivity. The strong -ELR was introduced and studied. Among others, Rosenthal’s Theorem (1.2.1) was strengthened and extended in as follows. Theorem (1.2.6) [1] (Oja). Let be an operator ideal and let be a Banach space. Let 1 ≤ < ∞ . If () ⊂ () and ∗ has the - bounded ()-approximation property with conjugate operators, then is strongly -extendably locally reflexive. It was also observed that Theorem (1.2.1) fails already for the bounded compact approximation property, i.e., ℱ() cannot be replaced by () in Theorem (1.2.1). This also means that the assumption “ has the -bounded ()-approximation property with conjugate operators” is essential in Theorem (1.2.6). A “strong” example of this phenomenon

6 was presented: there exists a strongly 1-ELR Banach space with a monotone shrinking basis such that: (i) its even duals ∗∗, ∗∗∗∗, ... are strongly 1-ELR, have the metric compact approximation property, but do not have the bounded weakly compact approximation property with conjugate operators; (ii) its odd duals ∗, ∗∗∗, ... are not ELR, but have the metric compact approximation property with conjugate operators. We shall extend Theorem (1.2.6) to convex approximation properties of pairs as follows. Theorem (1.2.7) [1] Let be a Banach space and a closed subspace of . Let be a linear subspace of ℒ() containing ℱ(). Let 1 ≤ < ∞. If ∗ has the -bounded { ∈ : () ⊂ }-approximation property with conjugate operators, then for every finite-dimensional subspace ⊂ ∗ and for every > 0, there exists an operator ∈ with () ⊂ such that ‖‖ ≤ + and ∗∗ = ∗ for all ∗ ∈ . Moreover, if ⊂ () , then the operator : = ∗∗ has the following properties: (∗∗) ⊂ , () ⊂ , ‖‖ ≤ + , ∗(∗∗) = ∗∗(∗) for all ∗∗ ∈ ∗∗ and ∗ ∈ , and ∗(∗) ⊂ ∗. In the proof of Theorem (1.2.7), we use the lemma below, which, by standard arguments, follows from the definition. Proof. Let ⊂ ∗ be a finite-dimensional subspace and let > 0. Look at ∗ endowed with its weak∗ topology and notice that is weak∗ closed. Using [1] choose a weak∗-to-weak∗ continuous linear projection on ∗ such that ran = and () ⊂ . Then there exists ∈ ℱ() such that = ∗. Hence, () ⊂ and = ran∗. Moreover, by assumption and Lemma (1.2.8), we have ∈ with () ⊂ such that ‖∗‖ ≤ and ‖∗∗ − ∗‖ ≤ (/‖‖)‖∗‖ for all ∗ ∈ .

Define = + ( − ) . Then, clearly, ∈ and () ⊂ . Let us observe that ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ‖( − ) ‖ = sup ‖ − ( )‖ ≤ sup ‖ ‖ ∗ ∗ ‖ ‖ ∈∗ ∈∗ = . ∗ ∗ ∗ Hence, ‖‖ ≤ ‖ ‖ + ‖(∗ − ) ‖ ≤ + . Let us also observe that ∗ ∗ ∗ = ∗ + (∗ − )( − ∗). Hence, clearly, ∗ is identity on = ran ∗.

Assume now that ⊂ (). Then ∈ () and ∗ ∈ (∗). Therefore, : = ∗∗ ∈ (∗∗, ) and ∗ ∈ (∗∗∗, ∗). Moreover, since () ⊂ , we get that () ⊂ ∩ = . And we also have ∗(∗∗) = ∗∗(∗∗) = ∗∗(∗) ∀∗∗ ∈ ∗∗, ∀∗ ∈ . We extend the strong ELR to pairs as follows. Lemma (1.2.8) [1] Let be a Banach space. Let be a linear subspace of ℒ() . Let 1 ≤ < ∞ . If has the -bounded -approximation property, then for every finite-dimensional subspace of and for every > 0 there exists an operator ∈ with ‖‖ ≤ , such that ‖ − ‖ ≤ ‖‖ for all ∈ . Definition (1.2.9) [1] Let be a Banach space and a closed subspace of . Let 1 ≤ < ∞. We say that the pair (, ) is strongly -extendably locally reflexiveif for all finite-dimensional subspaces ⊂ ∗∗ and ⊂ ∗ , and for all > 0 , there exists ∈ (∗∗) such that () ⊂ , () ⊂ , ‖‖ ≤ + , ∗(∗∗) = ∗∗(∗) for all ∗∗ ∈ and ∗ ∈ , and ∗(∗) ⊂ ∗. It is natural to say that the pair (∗, ) has the -bounded - approximation property with conjugate operators if ∗ has the -bounded { ∈ : () ⊂ } -approximation property with conjugate operators. Thus, the “moreover” part of Theorem (1.2.7) may be reformulated as follows. Theorem (1.2.10) [1] Let be a Banach space and a closed subspace of . Let be a linear subspace of () containing ℱ(). Let 1 ≤ < ∞. If the pair (∗, ) has the -bounded -approximation property with conjugate operators, then the pair (, ) is strongly -extendably locally reflexive. Theorem (1.2.10) contains Theorem (1.2.6) as the special case when = {0} and is the component of an arbitrary operator ideal, since the strong ELR of coincides with the strong ELR of the pair (, {0}). The theorem of Johnson, mentioned in the beginning of this section, was extended from ∗ to (∗, ): if the pair (∗, ) has the - bounded approximation property, then it has the -bounded approximation property with conjugate operators. Therefore, taking = ℱ(), we immediately get from Theorem (1.2.10) the following version of Rosenthal’s Theorem (1.2.1) for pairs.

Corollary (1.2.11) Let be a Banach space and a closed subspace of . Let 1 ≤ < ∞. If the pair (∗, ) has the -bounded approximation property, then the pair (, ) is strongly -extendably locally reflexive.

Chapter 2 Convex Cyclic Operators We give a Hahn- Banach characterization for convex-cyclic. We also obtain an example of abounded operator on a Banach space with ∗ ( ) = ∅ such that is convex- cyclic, but is not weakly hypercyclic and is not convex- cyclic. This solved two questions of Rezaci ∗ when ( ) = ∅. We also characterize the diagnolizable normal operators that are convex- cyclic and give a condition on the eigenvalues of an arbitrary operators for it to be convex- cyclic. Section (2.1): The Hahn-Banach Characterization for Convex- Cyclicity The convex hull of a given set may be defined as: (i) The (unique) minimal convex set containing ; (ii) The intersection of all convex set containing [6]. Rezaei gave a (universality) criterion for an operator to be convex- cyclic. Using the Hahn-Banach Separation Theorem, we give a necessary and sufficient condition for a set to have a dense convex hull, as a result we get a criterion for a vector to be a convex-cyclic vector for an operator. Proposition (2.1.1) [2] Let be a locally convex space over the real or complex numbers and let be a nonempty subset of . The following are equivalent: (i) The convex hull of is dense in . (ii) For every nonzero continuous linear functional on we have that the convex hull of Re() is dense in ℝ. (iii) For every nonzero continuous linear functional on we have that sup () = ∞ and inf () = −∞. (iv) For every nonzero continuous linear functional on we have that sup (()) = ∞. Proof: Let denote either the real or complex numbers. Clearly (i) ⇒ (ii) ⇒ (iii) ⇒ (iv) holds. Now assume that (iv) holds and by way of contradiction, assume that () is not dense in . Then there exists a point ∈ that is not in the closure of (). So, by the Hahn-Banach Separation Theorem, there exists a continuous linear functional on so that () < () for all ∈ (). It follows that () is

10 bounded from above and thus sup (()) ≠ ∞. This contradicts our assumption that (iv) is true. Thus it must be the case that if (iv) holds, then (i) does also. Hence all four conditions are equivalent. Corollary (2.1.2) [2] (The Hahn-Banach Characterization for Convex- Cyclicity). Let be a locally convex space over the real or complex numbers, : → a continuous linear operator, and ∈ . Then the following are equivalent: (i) The convex hull of the orbit of under is dense in . (ii) For every non-zero continuous linear functional on we have sup (, ) = ∞. Below are some simple consequences of the Hahn-Banach characterization for convex-cyclic vectors. As it was pointed the range of a cyclic operator may not be dense. For example, the range of the unilateral shift has codimension one. The closure of the range of a cyclic operator has codimension at most one. Notice that the range of hypercyclic operator is always dense. The Hahn- Banach characterization of convex-cyclicity easily shows that convex- cyclic operators must also have dense range, see the following result. Proposition (2.1.3) [2] If is a convex-cyclic operator on a locally convex space , then has dense range. Proof: Suppose that is a convex-cyclic operator and let be a convex- cyclic vector for , and by way of contradiction, suppose that does not have dense range. Then there exists a continuous linear functional such that () = {0}, where () denotes the range of . By the Hahn- Banach characterization, Corollary (2.1.2), we must have that sup ((, ) = ∞. However, since ∈ () for all ≥ it follows that () = 0 for all ≥ 1 . So, sup ((, ) = sup ({(0), 0}) < ∞. It follows from Corollary (2.1.2) that is not a convex-cyclic vector, a contradiction. Thus, must have dense range. In general, if is hypercyclic and > 1, then may not be hypercyclic. However, León-Saavedra and Müller proved that if is hypercyclic and is a unimodular complex number, then is hypercyclic. The same property is also true for weak hypercyclic operators. Next we present a similar result for convex-cyclic operators, that follows from the Hahn-Banach characterization of convex-cyclic vectors.

Proposition (2.1.4) [2] If is a convex-cyclic operator on a real or complex locally convex space , and if > 1, then is also convex- cyclic. Furthermore, every convex-cyclic vector for is also a convex- cyclic vector for . Proof: Suppose that is a convex-cyclic vector for , and we will show that is also a convex-cyclic vector for , by using the Hahn-Banach characterization (Corollary (2.1.2)). Let be any non-zero continuous linear functional on . Since is a convex-cyclic vector for , then sup (()) = ∞. Since > 1, then we have that sup [(())] = sup [()] ≥ sup [()] = ∞ . So, by the Hahn-Banach characterization, is a convex-cyclic vector for . Corollary (2.1.5) If || ≥ 1 and is weakly hypercyclic, then is convex-cyclic. Proof: Let : = , where ∈ ℝ and ≥ 1. Then by dela Rosa. We obtain that is weakly hypercyclic, hence is convex-cyclic. Thus, = is convex cyclic by Proposition (2.1.4). Let us define the following convex polynomials 1 + + ⋯ + = 1 (); = − 1 ( + + ⋯ + ) > 1. − 1 Definition (2.1.6) [2] Let and be topological spaces. A family of continuous operators : → ( ∈ ) is universal if there exists an ∈ such that {: ∈ } is dense in . Let ∈ (). Denotes () the arithmetic means given by + + ⋯ + () ≔ . Recall that an operator is Cesáro hypercyclic if there exists ∈ such that {(): ∈ ℕ} is dense in . See [2]. It is proved that is Cesáro hypercyclic if and only if is universal. Proposition (2.1.7) [2] Let be a Banach space, > 1 and ∈ () such that − has dense range. Then the following are equivalent: (i) is hypercyclic (ii) () is universal. ∈ℕ Proof: Notice that if > 1,

()( − ) = ( − )() = ( − 1) − . − 1 Proposition (2.1.8) [2] If is Cesáro hypercyclic or is hypercyclic for some ≥ 1, then is convex-cyclic. Notice that the proof of the sufficient condition for a bilateral weighted backward shift on ℓ(ℤ) to be convex-cyclic given is not correct. Section (2.2): -hypercyclic operators versus convex-cyclic operators with diagonal operators and adjoint multiplication operators

Let us now exhibit the relation between -hypercyclic and convex- cyclic operators. Theorem (2.2.1) [2] Every -hypercyclic vector is a convex-cyclic vector. Proof: Let be an -hypercyclic vector for an operator and we will prove that for a non-zero vector ∈ and > 0, there exists a convex polynomial such that ‖() − ‖ < . Since ∈ (0, 1), there exists ∈ ℕ such that 2‖‖ < . As is an - hypercyclic vector for , there exists a positive integer such that − ≤ ‖‖ = ‖‖. If − = 0, we choose such that − ≤ ‖‖. − 1 − 1 Thus − ≤ ‖‖. − 1 Hence 1 − 1 − 1 + − = ≤ 2 ‖‖ < and the proof ends by letting () = + . If − ≠ 0, there exists a positive integer such that + − = − ( − ) ≤ − ≤ ‖‖. If + − = 0, analogously to the above situation we choose such that

1 1 − 2 − 1 + + − = ≤ 2 ‖‖ < and the proof ends. If + − ≠ 0, there exists a positive integer such that + + − ≤ ‖‖. By induction, in the step , if + + ⋯ + − =

0, we choose such that 1 1 1 1 + + ⋯ + + − ≤ 2 ‖‖ < and the proof ends. If + + ⋯ + − ≠ 0, there exists a positive integer such that + + ⋯ + + − ≤ ‖‖ Thus + ⋯ + − ≤ ‖‖ < Ending completely the Proof: By a Fréchet space we mean a locally convex space that is complete with respect to a translation invariant metric. If is a nonempty collection of polynomials and is an operator on a space , then is said to be -cyclic and ∈ is said to be an - cyclic vector for if {(): ∈ } is dense in . Furthermore, is said to be -transitive if for any two nonempty open sets and in , there exists a ∈ such that () ∩ ≠ ∅ . Since the set of all polynomials with the topology of uniform convergence on compact sets in the complex plane forms a separable metric space, then any set of polynomials is also separable, hence we have the following result. Proposition (2.2.2) [2]. Suppose that : → is a continuous linear operator on a real or complex Fréchet space and is a nonempty set of polynomials. Then the following are equivalent: (i) has a dense set of -cyclic vectors. (ii) is -transitive. That is, for any two nonempty open sets , in , there is a polynomial ∈ such that () ∩ ≠ ∅.

(iii) has a dense set of -cyclic vectors.

By choosing various sets of polynomials for , we can get results for hypercyclic and supercyclic operators, as well as cyclic operators that have a dense set of cyclic vectors. If is the set of all convex polynomials, then we get the following immediate corollary. Corollary (2.2.3) [2]. Let : → be a continuous linear operator on a real or complex Fréchet space, then the following are equivalent. (i) has a dense set of convex-cyclic vectors. (ii) is convex-transitive. That is, for any two nonempty open sets , in , there is a convex polynomial such that () ∩ ≠ ∅.

(iii) has a dense set of convex-cyclic vectors. Proposition (2.2.4) [2] Let be a nonempty set of polynomials and let {: → } be a uniformly bounded sequence of linear operators on a sequence of Banach spaces {} such that for every ≥ 1, the () operator = ⨂ on = ⨂ has a dense set of -cyclic () vectors. Then = ⨂ is -cyclic on = ⨂ and has a dense set of -cyclic vectors.

Proof: Suppose that for every ≥ 1 the operators are -cyclic and have a dense set of -cyclic vectors. We will show that is -transitive. Let and be two nonempty open sets in (). Since the vectors in with only finitely many non-zero coordinates are dense in , then we may () choose vectors = () and = () in such that = 0 and = 0 for all sufficiently large , say = 0 and = 0 for all ≥ , and such that ∈ and ∈ . Since is -cyclic and has a dense set of -cyclic vectors in () , there exists a vector = () (, , … , ) ∈ such that u is an -cyclic vector for and so that (, , … , ) is close enough to (, , … , ) so that the infinite vector = (, , … , , 0,0, … ) ∈ . Since is -cyclic, there is a polynomial ∈ such that ()(, , … , ) is close enough to () (, , … , ) such that () ∈ . Thus, is -transitive on , and thus by Proposition (2.2.2) we have that has a dense set of - cyclic vectors. We next apply the previous proposition to infinite diagonal operators where is the set of all convex polynomials. This extends the finite dimensional matrix result given by Rezaei to infinite dimen-sional diagonal matrices.

Theorem (2.2.5) [2] Suppose that is a diagonalizable normal operator on a separable (real or complex) Hilbert space with eigenvalues {}. (a) If the Hilbert space is complex, then is convex-cyclic if and only if we have that the eigenvalues {} are distinct and for every ≥ 1, || > 1 and () ≠ 0. (b) If the Hilbert space is real, then is convex-cyclic if and only if the eigenvalues {} are distinct and for every ≥ 1 we have that < −1. Proof: By the spectral theorem we may assume that = (, , … ) is an infinite diagonal matrix acting on ℓℂ(ℕ) and let {} be the canonical unit vector basis where has a one in its coordinate and zeros elsewhere. (a) If is convex-cyclic with convex-cyclic vector = () ∈ ℓℂ(ℕ), then by Corollary (2.1.2) we must have for every ≥ 1 that ∞ = sup (〈 , 〉) = sup (λ). This implies that ≠ 0 and that || > 1 for each ≥ 1. Likewise, since the Hilbert space is complex in this case, we must have

− ∞ = sup 〈 , 〉 = sup λ = sup (λ) . This implies that cannot be real, hence () ≠ 0 for all ≥ 1. Conversely, suppose that for every ≥ 1 we have that || > 1 and () ≠ 0. Then for ≥ 1, let : = (, , … , ) be the diagonal matrix on ℂ where is the diagonal entry. Since the eigenvalues {} are distinct and || > 1 and () ≠ 0 for 1 ≤ ≤ , then we know from Rezaei that is convex-cyclic on ℂ and that every vector all of whose coordinates are non-zero is a convex-cyclic vector for . Since such vectors are dense in ℂ for every ≥ 1, then it follows from Proposition (2.2.4) that is also convex-cyclic and has a dense set of convex-cyclic vectors. (b) The proof of the real case is similar to that above. The next theorem says that if an operator has a complete set of eigenvectors whose eigenvalues are distinct, not real, and lie outside of the closed unit disk, then the operator is convex-cyclic. Theorem (2.2.6) [2] Let : : > 1 and 0 < || < = ℂ\ ( ∪ ℝ). Suppose that is a bounded linear operator on a complex Banach space and that has a countable linearly independent set of

16 eigenvectors with dense linear span in such that the corresponding eigenvalues are distinct and are contained in the set . Then is convex- cyclic and has a dense set of convex-cyclic vectors. Proof: Suppose that {} is a linearly independent set of eigenvectors for that have dense linear span in and such that the corresponding eigenvalues {} are distinct and contained in the set . By replacing each eigenvector with a constant multiple of itself we may assume that ∑‖‖ < ∞. Let be the diagonal normal matrix on ℓ (ℕ) whose diagonal entry is . Then define a linear map : ℓ (ℕ) → by ({}) = ∑ . Notice that since {} ∈ ℓ (ℕ), then we have that / / ‖({})‖ = ≤ || ‖‖ = ‖{}‖ℓ(ℕ) / where ≔ (∑‖‖ ) , which is finite. The above inequality implies that is a well defined continuous linear map from ℓ(ℕ) to . It follows that since the eigenvectors {} have dense linear span in , that has dense range. Also, if {} is the standard unit vector basis in ℓ (ℕ), then clearly () = for all ≥ 1 and thus intertwines with . To see this notice that () = () = = () = (). Thus () = () for all ≥ 1, thus = . Finally, since has distinct eigenvalues that all lie in the set , it follows from Proposition (2.2.5) that is convex-cyclic and has a dense set of convex- cyclic vectors. Since intertwines and and has dense range, then will map convex-cyclic vectors for to convex-cyclic vectors for . Thus, is convex-cyclic and has a dense set of convex-cyclic vectors. If is an open set in the complex plane, then by a reproducing kernel Hilbert space ℋ of analytic functions on we mean a vector space of analytic functions on that is complete with respect to a norm given by an inner product and such that point evaluations at all points in are continuous linear functionals on ℋ. Naturally we also require that = 0 in ℋ if and only if () = 0 for all ∈ . This is equivalent to the reproducing kernels having dense linear span in ℋ. Given such a space ℋ, a multiplier of ℋ is an analytic function on so that ∈ ℋ for every ∈ ℋ . In this case, the closed graph theorem implies that the multiplication operator : ℋ → ℋ is a bounded linear operator.

Corollary (2.2.7) [2] Suppose that is an open set in ℂ with components

{}∈ and ℋ is a reproducing kernel Hilbert space of analytic functions on , and that is a multiplier of ℋ . If is non-constant on every component of and () ∩ { ∈ ℂ: || > 1} ≠ ∅ for every ∈ , then ∗ the operator is convex-cyclic on ℋ and has a dense set of convex- cyclic vectors. ∗ Proof: We will show that the eigenvectors for with eigenvalues in the set = ℂ\( ∪ ℝ) have dense linear span in ℋ. It will then follow from ∗ Theorem (2.2.6) that is convex-cyclic. ∗ Every reproducing kernel for ℋ is an eigenvector for . In fact, ∗ if ∈ , then = (λ) , where denotes the reproducing kernel for ℋ at the point ∈ . By assumption, for every component of , is non-constant on , thus the set { ∈ : |()| > 1} is a nonempty open subset of . Also since is an open map on , cannot map the open set { ∈ : |()| > 1} into ℝ . Thus, for all ∈ , = { ∈ : |()| > 1 and () ∉ ℝ} is a nonempty open subset of . Let ∗ : = ⋃∈ . Then for every ∈ , is an eigenvector for with eigenvalue () which lies in = ℂ\( ∪ ℝ) . Since ∩ is a nonempty open set for every ∈ , then the corresponding reproducing kernels {: ∈ } have dense linear span in ℋ. Finally, since is non- constant on for each ∈ , we can choose a countable set , , in that has an accumulation point in in such a way that is one-to- one on , . Then the countable set is a set of , , , independent eigenvectors with dense linear span in ℋ and with distinct ∗ eigenvalues. It now follows from Theorem (2.2.6) that is convex- cyclic and has a dense set of convex-cyclic vectors. Next we give an example of a convex-cyclic operator that is not 1- weakly hypercyclic. ∗ Example (2.2.8) [2] Let be the adjoint of the multiplication operator associated to the multiplier (): = 2 + on () . The ∗ operator = 2 + , where is the unilateral backward shift, is not ∗ 1-weakly-hypercyclic, however is convex-cyclic by Corollary (2.2.7). The following result is true since powers of convex polynomials are also convex polynomials.

Proposition (2.2.9) [2] If is an operator on a Banach space and there exists a convex polynomial such that () is hypercyclic, then is convex-cyclic. By a region in ℂ we mean an open connected set in ℂ. In the following theorem, we consider the operator which is the adjoint of multiplication by , the independent variable. Theorem (2.2.10) [2] Suppose that is a bounded region in ℂ and ∩ {: || > 1} ≠ ∅. Suppose also that ℋ is a reproducing kernel Hilbert ∗ space of analytic functions on , then is convex-cyclic on ℋ. In fact, ∗ there exists a convex polynomial p such that ( ) is hypercyclic on ℋ. Proof: Choose ≥ 1 such that : = {: ∈ } satisfies ∩ { ∈ ℂ: () < 1} ≠ ∅. To see how to do this, choose a polar rectangle = : < < and < < such that ⊆ . Then simply choose a positive integer such that ( − ) > 2. Then ⊆ and will contain the annulus : < < , so certainly ∩ { ∈ ℂ: () < 1} ≠ ∅.

Now if 0 < ≤ 1, then the convex polynomial () = + (1 − ) maps the disk , onto the unit disk. Notice that the family of disks , : 0 < < 1 is the family of all disks that are centered on the negative real axis and pass through the point = 1. Thus it follows that { ∈ ℂ: () < 1} = ⋃ , . So we can choose an ∈ (0, 1) such that ∩ , ≠ ∅ . It follows that the polynomial () = ( ) is a convex polynomial and furthermore it satisfies () ∩ ≠ ∅. ∗ ∗ # ∗ Thus is hypercyclic on ℋ . However, = () where #() = (̅). Also, since is a convex polynomial, all of its coefficients # ∗ # ∗ ∗ are real, thus = . Thus, () = () = is hypercyclic on ℋ. We show an example of an operator that is convex-cyclic but no convex polynomial of the operator is hypercyclic. In other words, the operator is purely convexcyclic. Example (2.2.11) [2] Let {} and {} be two strictly decreasing sequences of positive numbers that are interlaced and converging to zero.

In other words, 0 < < < for all ≥ 1 and → 0 (and hence → 0). For each ≥ 1, let

1 : = : 2 < < 2 + and < < . Let : = ⋃ and let () be the Bergman space of all analytic functions on that are square integrable with respect to area measure on ∗ . Then the operator is purely convex- cyclic on (); meaning that ∗ ∗ is convex-cyclic on (), but () is not hypercyclic on () for any convex polynomial . ∗ Proof: By Corollary (2.2.7) we know that is convex-cyclic on (). ∗ In order to show that no convex polynomial of is hypercyclic, suppose, by way of contradiction, that there exists a convex polynomial ∗ such that () is hypercyclic. Since is a convex polynomial it has real # # ∗ coefficients thus () = () where (): = (̅) . Thus () = ∗ ∗ ∗ # = and it follows that is hypercyclic on () . Thus it follows that every component of must satisfy that () ∩ ≠ ∅. However since is a convex polynomial, is (strictly) increasing on the interval [0, ∞). Thus, (2) > (1) = 1. Choose an > 0 such that < (2) − 1. Since is continuous at = 2, and since we have an > 0, then there exists a > 0 such that if – 2 < , then |() − (2)| < .

Notice that for sufficiently large we have that ⊆ (2, ) , thus, () ⊆ ((2), ) ⊆ { ∈ ℂ: () > 1} . Thus, () ∩ ≠ ∅ for ∗ all large , a contradiction. It follows that no convex polynomial of is ∗ hypercyclic, hence is purely convex-cyclic.

Chapter 3 Separably Injective Banach Spaces We obtain two fundamental characterization of universally separably injective spaces: (i) A Banach space is universally separably injective if and only if every separable subspaces is contained in a copy of

ℓ inside . (ii) A Banach space is universally separably injective if ℓ and only if for every separable space on has Ext , = 0. We construct a consistent example of a Banach space of type () which is 1-separably injective but not 1-universally separably injective. Section (3.1): Basic properties of separably injective spaces with examples A Banach space is separably injective if for every separable Banach space and each separable ⊂ , every operator : → extends to an operator : → . If some extension exists with ‖‖ ≤ ‖‖ we say that is -separably injective [7]. Definition (3.1.1) [3] A Banach space is separably injective if for every separable Banach space and each subspace ⊂ , every operator : → extends to an operator : → . If some extension exists with ‖‖ ≤ ‖‖ we say that is -separably injective. Every separably injective space is -separably injective for some

since every sequence of norm-one operators : → induces a norm-one operator : ℓ() → . Separable injective spaces can be characterized as follows. Proposition (3.1.2) [3] For a Banach space the following properties are equivalent. (a) is separably injective.

(b) Every operator from a subspace of ℓ into E extends to ℓ. (c) For every Banach space and each subspace such that / is separable, every operator : → extends to . (d) If is a Banach space containing and / is separable, then is complemented in . (e) For every separable space one has Ext (, ) = 0. Moreover, (i) The space is -complemented in every such that / is separable if and only if every operator : → admits an extension : → with ‖‖ ≤ ‖‖, whenever / is separable.

(ii) If is -separably injective, then for every operator : → there exists an extension : → of with ‖‖ ≤ 3‖‖, whenever / is separable. Proof: It is clear that (c) ⇒ (a) ⇒ (b) and (c) ⇒ (d) ⇔ (e). Moreover, (i) shows that (d) ⇒ (c) and (ii) shows that (a) ⇒ (c). The remaining implication (b) ⇒ (a) follows from the proof of (ii) below. For the sufficiency statement in (i) simply consider as the identity on . For the necessity statement, given an operator : → form the associated push-out diagram 0 → → → / → 0 ↓ ↓ || 0 → → → / → 0. Since / = / is separable, there is a projection : → with norm at most , and thus, recalling that ‖‖ ≤ 1 , the composition : → yields an extension of with norm at most . The proof for (ii) is a little more tricky. Let be a surjective map from ℓ → /. The lifting property of ℓ provides an operator : ℓ → . Consider thus the commutative diagram 0 → ker → ℓ → / → 0 ↓ ↓ || 0 → → → / → 0 Let us construct the true push-out of the couple (, ) and the corresponding complete diagram 0 → ker → ℓ → / → 0 ↓ ↓ || 0 → → → / → 0.

We can consider without loss of generality that ‖‖ = 1. Let : ℓ → be an extension of with ‖‖ ≤ ‖‖ ≤ ‖‖ . By the universal property of the push-out, there exists an operator : → such that = and ‖‖ ≤ max{‖‖, ‖‖} ≤ ‖‖ . Again by the universal property of the push-out, there is a diagram of equivalent exact sequences 0 → → → / → 0 || ↓ || 0 → → → / → 0,

22 where the isomorphism is defined as (, ) + = () + () is such that ‖‖ ≤ max{‖‖, ‖‖} ≤ 1. The desired extension of to is = , where comes defined by () = − (), () + , where : / → ℓ is a homogeneous bounded selection for with ‖‖ ≤ 1. One clearly has ‖‖ ≤ 3, and therefore ‖‖ ≤ 3. We are especially interested in the following subclass of separably injective spaces. Definition (3.1.3) [3] A Banach space is said to be universally separably injective if for every Banach space and each separable subspace ⊂ , every operator : → extends to an operator : → . If some extension exists with ‖‖ ≤ ‖‖ we say that is universally -separably injective. A Banach space is universally separably injective if and only if every -valued operator with separable range extends to any superspace. It is also easy to show that every universally separably injective space is -universally separably injective for some . Recall that a Banach space has Pełczyński’s property () if each operator defined on is either weakly compact or it is an isomorphism on a subspace isomorphic to . We will say that has Rosenthal’s property () if it satisfies the preceding condition with ℓ replacing . It is well-known that Lindenstrauss spaces (i.e., ℒ -spaces) have this property.

Not all ℒ -spaces have Pełczyński’s property (): for example, the ℒ -spaces without copies of constructed by Bourgain and Delbaen; or those that can be obtained from Bourgain-Pisier; or the space Ω constructed as a twisted sum

0 → [0, 1] → Ω → → 0 with strictly singular quotient map. Recall that a Banach space is said to be a Grothendieck space if every operator from to a separable

Banach space (or to ) is weakly compact. Clearly, a Banach space with property () is a Grothendieck space if and only if it has no complemented subspace isomorphic to . It is well-known that ℓ is a Grothendieck space.

Proposition (3.1.4) [3]

(a) A separably injective space is of type ℒ , has Pełczyński’s property () and, when it is infinite dimensional, contains copies

of . (b) A universally separably injective space is a Grothendieck space of

type ℒ , has Rosenthal’s property () and, when it is infinite dimensional, contains ℓ. Proof: (a) Let be a -separably injective space. We want to see that if is a subspace of any Banach space , every operator : → extends to an operator : → ∗∗ with ‖‖ ≤ ‖‖. This implies that ∗∗ is - injective, by an old result of Lindenstrauss. Being of infinite dimension, ∗∗ is an ℒ, space and so is . Let : → be an operator. Given a finite-dimensional subspace of , let : → be any operator extending the restriction of to ∩ . Let ℱ be the set of finite- dimensional subspaces of , ordered by inclusion, let be any ultrafilter refining the Fréchet filter on ℱ, that is, containing every set of the form { ∈ ℱ: ⊂ } for fixed ∈ ℱ. Then, define : → ∗∗ taking ∗ () = − lim 1() . () It is easily seen that is a linear extension of , with ‖‖ ≤ ‖‖.

To show that contains and has property (), let : → be a non-weakly compact operator ( being an infinite dimensional ℒ space cannot be reflexive). Choose a bounded sequence () in such that

has no weakly convergent subsequences and let be the subspace spanned by () in . As is separable we can regard it as a subspace of [0, 1]. Let : [0, 1] → be any operator extending the inclusion of into . Since : [0, 1] → is not weakly compact, is an isomorphism on some subspace isomorphic to ; and the same occurs to . (b) If, in addition to that, is universally separably injective we may take : → and ⊂ as before but this time we consider as a subspace of ℓ. If : ℓ → is any extension of the inclusion of into , then : ℓ → is not weakly compact. Hence it is an isomorphism on some subspace isomorphic to ℓ∞ and so is . Several modifications on the proof of Ostrovskii yield Proposition (3.1.5) A -separably injective space with < 2 is either finite-dimensional or has density character at least .

Recall that a class of Banach spaces is said to have the 3-space property if whenever / and belong to the class, then so does. Proposition (3.1.6) [3] (i) The class of separably injective spaces has the 3-space property. (ii) The quotient of two separably injective spaces is separably injective. (iii) The class of universally separably injective spaces has the 3- space property. (iv) The quotient of a universally separably injective space by a separably injective space is universally separably injective. Proof: The simplest proof for the 3-space property (i) follows from characterization (ii) in Proposition (3.1.2): let us consider an exact sequence 0 → → → → 0 in which both and are separably injective. Let : → be an operator from a subspace : → ℓ of ℓ; then can be extended to an operator Φ: ℓ → , which can in turn be lifted to an operator Ψ: ℓ → . The difference − Ψ takes values in and can thus be extended to an operator : ℓ → . The desired operator is Ψ + . A different homological proof that properties having the form Ext(, −) = 0 are always 3-space properties can be found. To prove (ii) and (iv) let us consider an exact sequence 0 → → → → 0 in which is separably injective and is (universally) separably injective. Let : → be an operator from a separable space which is a subspace of a separable (arbitrary) space . Consider the pull-back diagram 0 → → → → 0 || ↑ ↑ 0 → → → → 0 Since is separably injective, the lower exact sequence splits, so has a selection operator : → . By the injectivity assumption about , there exists an operator : → agreeing with on . Then : → is the desired extension of . The proof for (iii) has to wait until Theorem (3.2.1) when a suitable characterization of universally separably injective spaces will be presented. Several variations of these results can be seen. It is obvious that if

()∈ is a family of -separably injective Banach spaces, then ℓ(, )

25 is -separably injective. The non-obvious fact that also (, ) is separably injective can be considered as a vector valued version of Sobczyk’s theorem. Proofs for this result have been obtained by Johnson- Oikhberg, Rosenthal, Cabello and Castillo-Moreno, each with its own estimate for the constant. These are 2 (implicitly), (1 + ), (3) and 6, respectively. Examples (3.1.7) [3]: All injective spaces are universally separably injective. Sobczyk theorem states that –and (Γ), in general– are 2-separably injective in its natural supremum norm. They are not universally separably injective since they do not contain ℓ. (a) Twisted sums. The 3-space property yields that twisted sums of separably injective are also separably injective. In particular:

(i) Twisted sums of and (Γ) : This includes the Johnson- Lindenstrauss spaces (ℳ) obtained taking the closure of the linear span in ℓ of the characteristic functions {1}∈ℕ and 1 for an uncountable almost disjoint family { } of ∈ ∈ subsets of ℕ . Marciszewski and Pol answer a question of Koszmider showing that there exist 2 almost disjoint families

ℳ generating non-isomorphic (ℳ)-spaces. (ii) Twisted sums of two nonseparable (Γ)spaces. This includes variations of the previous construction using the Sierpinski- Tarski generalization of the construction of almost disjoint families; the Ciesielski-Pol space; the WCG nontrivial twisted

sums of (Γ) obtained independently by Argyros, Castillo, Granero, Jimenez and Moreno and by Marciszewski .

(iii) Twisted sums of and ℓ, as those constructed. ℓ (iv) A twisted sum of and that is not complemented in any ()-space, as the one obtained. (b) The space ℓ(Γ): A typical 1-universally separably injective space is the space ℓ(Γ) of countably supported bounded functions : → ℝ, where Γ is an uncountable set. This space is isomorphic but not isometric to some () space, showing in this way that the theory of universally separably injective spaces does not run parallel with that of injective spaces. What makes this space universally separably injective space is the following property:

Definition (3.1.8) [3] We say that a Banach space is ℓ -upper- saturated if every separable subspace of is contained in some

(isomorphic) copy of ℓ inside . It is clear that an ℓ-upper-saturated space is universally separably injective. We will prove later that the converse also holds.

The space ℓ/ : Since ℓ is injective and is separably injective, it follows from Proposition (3.1.6) that ℓ/ is universally separably injective, although the constant is not optimal. It follows from

Proposition (3.1.13) (a) that ℓ/ is 1-universally separably injective, hence, it is ℓ- upper-saturated. This can be improved to show that every separable subspace of ℓ/ is contained in a subalgebra of ℓ/ isometrically isomorphic to ℓ. It is well-known that ℓ/ is not injective. The simplest proof appears in Rosenthal: an injective space containing () must also contain ℓ() ; it is well-known that ℓ/ contains () for || = while it cannot contain ℓ(). The proof is quite rough in a sense: it says that ℓ/ is uncomplemented in its bidual, a huge superspace. Denoting ℕ∗ = ℕ\ℕ , Amir had shown that (ℕ∗) is not complemented in ℓ(2 ), which provides another proof that / is not injective. Amir’s proof can be refined in order to get (ℕ∗) uncomplemented in a much smaller space. It can be shown that (ℕ∗) contains an uncomplemented copy of itself. Proposition (3.1.9) [3]. A () space is 1-separably injective if and only if is an -space. Simple examples show that when a ()-space is only isomorphic to a 1-separably injective then does not need to be an -space. It is an immediate consequence of Tietze’s extension theorem that a closed subset of an -space is an -space. In particular, ℕ∗ = ℕ\ℕ is an - space. Given a compact space , we write for its derived set, that is, the set of its accumulation points. This process can be iterated to define () as () with () = . We say that has height if () = . We say that has finite height if it has height for some ∈ ℕ. Proposition (3.1.10) [3] If is a compact space of height , then () is (2 − 1)-separably injective. Consequently, if is a compact space of

27 finite height then () is separably injective although it is not universally separably injective. Proof: Let ⊂ with separable and let : → () be a norm one operator. The range of is separable and every separable subspace of a () is contained in an isometric copy of (), where is the quotient of after identifying and when () = () for all ∈ . This is metrizable because is separable. Moreover, if has height , then has height at most and so it is homeomorphic to [0, ⋅ ] with < , < . Since [0, ⋅ ] is (2 + 1) -separably injective, our operator can be extended to an operator : → () with norm ‖‖ ≤ (2 + 1)‖‖ ≤ (2 − 1)‖‖, concluding the Proof: When is a metrizable compact of finite height , Baker showed that 2 − 1 is the best constant for separable injectivity, using arguments from Amir. There are some difficulties in generalizing those arguments for nonmetrizable compact spaces, so we do not know if it could exist a nonmetrizable compact space of height such that () is -separably injective for some < 2 − 1. Proposition (3.1.11) [3] The space of all bounded Borel (respectively, Lebesgue) measurable functions on the line is 1-separably injective in the sup norm. Proof: Clearly the given spaces are in fact Banach algebras satisfying the inequality required by Albiac- Kalton characterization. Thus they can be represented as () spaces. On the other hand, each measurable function can be decomposed as = ||, with measurable. This clearly implies that the corresponding compacta are -spaces. Argyros proved that none of the spaces in the above example is injective. This is very simple in the Borel case: the characteristic functions of the singletons generate a copy of (ℝ) in the space of bounded Borel functions. The density character of the latter space is the continuum, as there are c Borel subsets. Therefore it cannot contain a copy of ℓ(ℝ), whose density character is 2 . (c) -ideals. A closed subspace ⊂ is called an -ideal if its annihilator = {∗ ∈ ∗: 〈∗, 〉 = 0 ∀ ∈ } is an - summand in ∗. This just means that there is a linear projection on ∗ whose range is and such that ‖∗‖ = ‖(∗)‖ + ‖∗ − (∗)‖ for all ∗ ∈ ∗.

The easier examples of -ideals are just ideals in () -spaces. In particular, if is a closed subset of the compact space and = \ one has that () is an -ideal in () is straightforward from the Riesz representation of ()∗. A remarkable generalization of Borsuk- Dugundji theorem for -ideals was provided by Ando and, independently, Choi and Effros. In order to state it let us recall that a Banach space has the -approximation property (-AP, for short) if, for every > 0 and every compact subset of , there exists a finite rank operator on , with ‖‖ ≤ , such that ‖ − ‖ < , for every ∈ . We say that has the bounded approximation property (BAP for short) if it has the -AP, for some . Theorem (3.1.12) [3]. Let be an -ideal in the Banach space and : → / the natural quotient map. Let be a separable Banach space and : → / be an operator. Assume further that one of the following conditions is satisfied: (i) has the -AP. (ii) is a Lindenstrauss space. Then can be lifted to , that is, there is an operator : → such that = . Moreover one can get ‖‖ ≤ ‖‖ under the assumption (i) and ‖‖ = ‖‖ under (ii). One has. Proposition (3.1.13) [3] Let be an -ideal in a Banach space . (a) If is -(universally) separably injective, then / is - (universally) separably injective. (b) If is -separably injective, then is 2 -separably injective. When is a Lindenstrauss space (which is always the case if is), then the exponent 2 disappears.

In particular, if is a closed subset of the compact space and = \ one has: (c) If () is -(universally) separably injective, then so is

(). (d) If () is -separably injective, then () is 2-separably injective. Proof: (a) By (the proof of) Proposition (3.1.4), ∗∗ is -injective and so ∗∗ ∗∗ ∗∗ ∗∗ it has the -AP. Since = ⊕ (/) we see that also and (/)∗∗ have the -AP. Hence both and (/) have the -AP. Let be

29 a separable subspace of and : → / an operator. Let be a separable subspace of / containing the image of . We may assume has the -AP. Let : → be the lifting provided by Theorem (3.1.12), so that ‖‖ ≤ . Now, if : → is an extension of , then : → / is an extension of , and this can be achieved with ‖‖ = ‖‖ ≤ ‖‖. (d) –and (b)–. Let us remark that if is a subspace of () containing () and /() is separable, then there is a projection : → () of norm at most 2. Indeed, /() is a separable subspace of () and there is a lifting : /() → (), with ‖‖ = 1, and = − is the required projection. Now, let : → () be an operator, where is a subspace of a separable Banach space . Considering as taking values in (), there is an extension : → () with ‖‖ ≤ ‖‖ . Let denote the least closed subspace of

( ) containing the range of and () and : → () a projection with ‖‖ ≤ 2 . The composition : → () is an extension of t and clearly, ‖‖ ≤ 2‖‖.

(d) Ultraproducts of type ℒ : Let us briefly recall the definition and some basic properties of ultraproducts of Banach spaces. For a detailed study of this construction at the elementary level needed here we refer to Heinrich or Sims’ notes. Let be a set, be an

ultrafilter on , and ( )∈ a family of Banach spaces. Then ℓ() endowed with the supremum norm, is a Banach space, and ( ) () ∈ ℓ(): lim‖‖ = 0 is a closed subspace of () ℓ() . The ultraproduct of the spaces ( )∈ following is defined as the quotient ℓ() [ ] = . () We denote by [()] the element of [] which has the family () as a representative. It is not difficult to show that ‖[()]‖ = lim‖‖. In the () case = for all , we denote the ultraproduct by , and call it the ultrapower of following . If : → is a uniformly bounded family of operators, the ultraproduct operator []: [] → [] is given by [] [()] = [()]. Quite clearly, ‖[]‖ = lim‖‖. ()

Definition (3.1.14) [3] An ultrafilter on a set is countably incomplete if there is a decreasing sequence () of subsets of such that ∈ for all , and ⋂ = ∅. Notice that is countably incomplete if and only if there is a function : → ℕ such that () → ∞ along (equivalently, there is a family () of strictly positive numbers converging to zero along ). It is obvious that any countably incomplete ultrafilter is non-principal and also that every non-principal (or free) ultrafilter on ℕ is countably incomplete. Assuming all free ultrafilters countably incomplete is consistent with ZFC, since the cardinal of a set supporting a free countably complete ultrafilter should be measurable, hence strongly inaccessible.

It is clear that the classes of ℒ, spaces are stable under ultraproducts. In the opposite direction, a Banach space is a ℒ, space if and only if some (or every) ultrapower is. In particular, a Banach space is an ℒ space or a Lindenstrauss space if and only if so are its ultra powers. It is possible however to produce a Lindenstrauss space out of non-even-ℒ-spaces: indeed, if () → ∞ along , then the ultraproduct is a Lindenstrauss space. () The following result about the structure of separable subspaces of ultraproducts of type ℒ will be fundamental. Lemma (3.1.15) [3]: Suppose [] is an ℒ, -space. Then each separable subspace of [] is contained in a subspace of the form [], () where ⊂ is finite dimensional and lim , ℓ ≤ λ, where ()

() = dim . Proof: Let us assume is an infinite-dimensional separable subspace of [ ] . Let ( ) be a linearly independent sequence spanning a dense subspace in and, for each , let ( ) be a fixed representative of in ℓ( ). Let = span { , … , }. Since [] is an ℒ, -space there is, for each , a finite dimensional ⊂ [ ] containing with , ℓ ≤ + 1/. For fixed , let () be a basis for containing , … , . ℓ ℓ Choose representatives ( ) such that = if = . Moreover, let be the subspace of spanned by for 1 ≤ ≤ dim . Let () be a decreasing sequence of subsets ∈ such that ⋂ = ∅. For each integer put

2 = ∈ : , ℓ ≤ + ∩ and = ⋂ . All these sets are in . Finally, set = . Next we define a function : → ℕ. Set 1 ∈ () = sup{: ∈ } ∉ () For each ∈ , take = . This is a finite-dimensional subspace of whose Banach-Mazur distance to the corresponding ℓ is at most + 2/(). It is clear that [] contains and also that () → ∞ along , which completes the Proof:

Theorem (3.1.16) [3] Let ()∈ be a family of Banach spaces such that [ ] is a ℒ,-space. Then [ ] is -universally separably injective. () Lemma (3.1.17) [3] For every function : → ℕ, the space ℓ is 1- universally separably injective. Proof: Let Γ be the disjoint union of the sets {1, 2, … , ()} viewed as a () () discrete set. Now observe that ℓ is an ideal in ℓ ℓ =

ℓ(Γ) = (Γ) and apply Proposition (3.1.13) (a). Corollary (3.1.18) [3] Let ()∈ be a family of Banach spaces. If [ ] is a Lindenstrauss space, then it is 1-universally separably injective.

Section (3.2): Two characterizations of universally and 1-separably injective spaces In Proposition (3.1.4) (b) it was proved that universally separably injective spaces contain ℓ. Much more is indeed true: Theorem (3.2.1) [3] An infinite-dimensional Banach space is universally separably injective if and only if it is ℓ-upper-saturated. Proof: The sufficiency is a consequence of the injectivity of ℓ. In order to show the necessity, let be a separable subspace of a universally separably injective space . We consider a subspace of ℓ isomorphic to and an isomorphism : → . We can find projections on and on ℓ such that ⊂ ker , ⊂ ker , and both and have range isomorphic to ℓ. Indeed, let : → / be the quotient map. Since contains ℓ and is separable, is not weakly compact so, by Proposition (3.1.4) (b), there exists a subspace of isomorphic to ℓ where is an isomorphism. Now / = () ⊕ , with a closed subspace. Hence = ⊕ (), and it is enough to take as the projection with range and kernel (). Since ker and ker are universally separably injective spaces, we can take operators : → ker and : ℓ → ker such that = on and = on . Let : ℓ → ran be an operator satisfying ‖()‖ ≥ ‖‖ for all ∈ ℓ. We will show that the operator

= + 1ℓ − : ℓ → is an isomorphism (into). This suffices to end the proof since ran is isomorphic to ℓ and both and agree with t on , so ⊂ ran ⊂ . Since ran ⊂ ker and ran ⊂ ran , there exists > 0 such that ‖ ‖ ‖ ( )‖ ( ) ≥ max , 1ℓ − ∈ ℓ . Now, if ‖‖ < (2‖‖)‖‖, then ‖‖ < ‖‖; hence 1 1 − ≥ 1 − > ‖‖. ℓ ℓ 2 Thus ‖‖ ≥ (2‖‖)‖‖ for every ∈ . We can now complete the proof of Proposition (3.1.6) (iii) and show that the class of universally separably injective spaces has the 3- space property.

Proposition (3.2.2) [3] The class of universally separably injective spaces has the 3-space property.

Proof: By Theorem (3.2.1) one has to show that being ℓ -upper- saturated is a 3-space property. Let 0 → → → → 0 be an exact sequence in which both , are ℓ-uppersaturated, and let be a separable subspace of . It is not hard to find separable subspaces , of such that ⊂ and / = [()]. Let be a copy of ℓ inside containing . By the injectivity of ℓ, is contained in the subspace ⊕ [()] of . And since there exists a copy of ℓ containing [()], is therefore contained in the subspace ⊕ of , which is isomorphic to ℓ. A homological characterization of universally separably injective spaces is also possible. We need first to show: Proposition (3.2.3) [3]. If is a universally separably injective space then Ext (ℓ, ) = 0. Proof: James’s well known distortion theorem for ℓ(resp. ) asserts that a Banach space containing a copy of ℓ(resp. ) also contains an almost isometric copy of ℓ(resp. ). Not so well known is Partington’s distortion theorem for ℓ : a Banach space containing ℓ contains an almost isometric copy of ℓ (see also Dowling). This last copy will therefore be, say, 2-complemented.

Let Γ denote the set of all the 2-isomorphic copies of ℓ inside ℓ. For each ∈ Γ let : → ℓ be the canonical embedding, a projection onto of norm at most 2 and : → ℓ an isomorphism. Assume that a nontrivial exact sequence

0 → → → ℓ → 0 exists. We consider, for each ∈ Γ, a copy of the preceding sequence, and form the product of all these copies 0 → ℓ(Γ, ) → ℓ(Γ, ) → ℓ(Γ, ℓ) → 0 . Let us consider the embedding : ℓ → ℓ(Γ, ℓ) defined as ()() = () and then form the pull-back sequence 0 → ℓ(Γ, ) → ℓ(Γ, ) → ℓ(Γ, ℓ) → 0 || ↑ ↑ 0 → ℓ(Γ, ) → PB → ℓ → 0 Let us show that cannot be an isomorphism on a copy of ℓ . Otherwise, it would be an isomorphism on some ∈ and thus the new pull-back sequence

0 → ℓ(Γ, ) → PB → ℓ → 0 || ↑ ↑ 0 → ℓ(Γ, ) → PB → → 0 would split. And therefore the same would be true making push-out with the canonical projection : ℓ(Γ, ) → onto the -th copy of : 0 → ℓ(Γ, ) → PB → → 0 ↓ ↓ || 0 → → PO → → 0 But it is not hard to see that new pull-back with 0 → → PO → → 0 || ↑ ↑ 0 → → → ℓ → 0 produces exactly the starting sequence which, by assumption, was nontrivial. However, the space PB should be universally separably injective by Proposition (3.1.6) (iii), hence it must have Rosenthal’s property (), by Proposition (3.1.4) (b). This contradiction shows that the starting nontrivial sequence cannot exist. We are thus ready to prove: Theorem (3.2.4) [3] A Banach space is universally separably injective if and only if for every separable space one has Ext (ℓ/, ) = 0. Proof: Let be separable and let be universally separably injective.

Applying (−, ) to the sequence 0 → → ℓ → ℓ/ → 0 one gets the exact sequence

… → (ℓ, ) → (, ) → Ext(ℓ/, ) → Ext(ℓ, ) Since Ext(ℓ, ) = 0, one obtains that every exact sequence 0 → → → ℓ/ → 0 fits in a push-out diagram 0 → → ℓ → ℓ/ → 0 ↓ ↓ || 0 → → → ℓ/ → 0. Since is universally separably injective, the lower sequence splits. The converse is clear: every operator : → from a separable Banach space into a space produces a push-out diagram 0 → → ℓ → ℓ/ → 0 ↓ ↓ || 0 → → PO → ℓ/ → 0.

The lower sequence splits by the assumption Ext(ℓ/, ) = 0 and so extends to ℓ, according to the splitting criterion for push-out sequences. Which leads to the unexpected:

Corollary (3.2.5) [3] Ext(ℓ/, ℓ/) = 0 ; i.e., every short exact sequence 0 → ℓ/ → → ℓ/ → 0 splits. This result provides a new solution for equation Ext(, ) = 0.

The other three previously known types of solutions are: (by Sobczyk theorem), the injective spaces (by the very definition) and the ()- spaces (by Lindenstrauss’ lifting). Also with Proposition (3.2.2), one has: Corollary (3.2.6) [3] Rosenthal’s property () is not a 3-space property Proof: With the same construction as above, start with a nontrivial exact sequence 0 → ℓ → → ℓ → 0 (see [3]) and construct an exact sequence 0 → ℓ(Γ, ℓ) → → ℓ → 0, where cannot be an isomorphism on a copy of ℓ . So has not Rosenthal’s property (). The space ℓ(Γ, ℓ) has Rosenthal’s property () as a quotient of ℓ(Γ, ℓ) = ℓ(ℕ × Γ) , since the property obviously passes to quotients. It is not however true that Ext(, ) = 0 for all universally separably injective spaces and as any exact sequence 0 → →

ℓ(Γ) → ℓ(Γ)/ → 0 in which is a universally separably injective non-injective space shows. We establishe a major difference between 1-separably injective and general separably injective spaces: 1-separably injective spaces must be Grothendieck (hence they cannot be separable or WCG) while a 2- separably injective space, such as , can be even separable. The following lemma due to Lindenstrauss provides a quite useful technique. Lemma (3.2.7) [3] Let be a 1-separably injective space and a separable subspace of , with dens = ℵ. Then every operator : → can be extended to an operator : → with the same norm. This yields Proposition (3.2.8) [3] Under CH every 1-separably injective Banach space is universally 1-separably injective and therefore a Grothendieck space. Proof: Let be 1-separably injective, an arbitrary Banach space and : → an operator, where is a separable subspace of . Let [()] be

36 the closure of the image of . This is a separable subspace of and so there is an isometric embedding : [()] → ℓ . As ℓ is 1-injective there is an operator : → ℓ whose restriction to agrees with . Thus it suffices to extend the inclusion of [()] into to ℓ. But, under CH, the density character of ℓ is ℵ and the preceding Lemma applies. The ‘therefore’ part is now a consequence of Proposition (3.1.4) (b). The “therefore” part survives in ZFC: Theorem (3.2.9) [3] Every 1-separably injective space is a Grothendieck and a Lindenstrauss space. Proof: The proof of Proposition (3.1.4) yields that 1-separably injective spaces are of type ℒ, , that is, Lindenstrauss spaces. It remains to prove that a 1-separably injective space must be Grothendieck. It suffices to show that is not complemented in , so let : → be an embedding. Consider an almost-disjoint family ℳ of size ℵ formed by infinite subsets of ℕ and construct the associated Johnson-Lindenstrauss twisted sum space

0 → → (ℳ) → (ℵ) → 0. Observe that the space (ℳ) has density character ℵ , we have therefore a commutative diagram

0 → → (ℳ) → (ℵ) → 0 || ↓ ↓ 0 → → → /() → 0. If was complemented in then it would be complemented in (ℳ) as well, which is not. Proposition (3.2.8) leads to the question about the necessity of the hypothesis CH. We will prove now that it cannot be dropped. Lemma (3.2.10) [3] Let , , be compact spaces and let : → be a continuous map, with ℐ = ∘: () → () its induced operator, and let : () → () be a positive norm one operator. Suppose that : () → () is an operator with ‖‖ = 1 and = . Then is a positive operator. ∗ Proof: Obviously ≥ 0 if and only if ≥ 0 for all ∈ , where is the unit mass at and ∗: ()∗ → ()∗ is the adjoint operator. Fix ∗ ∈ . By Riesz theorem we have that = is a measure of total variation ‖‖ ≤ 1. Let = − be the Hahn-Jordan decomposition

of , so that ‖‖ = ‖ ‖ + ‖ ‖, with , ≥ 0. We have that () = ∗ ∗ ∗ ∗ = = , thus ∗ ∗ ∗ ∗ () = − and () = ‖ ‖ − ‖ ‖. Since is a positive operator these imply that the above is the Hahn- ∗ Jordan decomposition of () and so = 0, hence = 0. Definition (3.2.11) [3] Let be a zero-dimensional compact space. An

ℵ-Lusin family on is a family ℱ of pairwise disjoint nonempty clopen subsets of with |ℱ| = ℵ, such that whenever and ℋ are subfamilies of ℱ with || = |ℋ| = ℵ, then { ∈ } ∩ { ∈ ℋ} ≠ . The following lemma shows the consistency of the existence of an ∗ ℵ-Lusin family on ℕ . Lemma (3.2.12) [3] Under MA and the assumption = ℵ there exists an ∗ ℵ-Lusin family on ℕ . Proof: By Stone duality, since the Boolean algebra associated to ℕ∗ is ∗ ℘(ℕ)/fin, an ℵ-Lusin family on ℕ is all the same as an almost disjoint { } family of infinite subsets of ℕ such that for every ⊂ ℕ either {: |/| is finite} or {: | ∩ | is finite} has cardinality < ℵ . Let {: < } be an enumeration of all infinite subsets of ℕ. We construct the sets inductively on . Suppose has been constructed for < . We define a forcing notion ℙ whose conditions are pairs =

, where is a {0, 1}-valued function on a finite subset dom of ℕ and is a finite subset of . The order relation is that < if extends , ⊃ and vanishes in \ dom for ∈ . One checks that this forcing is . Hence, by MA, using a big enough generic filter the forcing provides an infinite set ⊂ ℕ such that, for all < , (i) ∩ is finite, and

(ii) If is not contained in any finite union of ’s, then ∩ is infinite. Theorem (3.2.13) [3] It is consistent that there exists a compact space for which the Banach space () is 1-separably injective but not universally 1-separably injective.

Proof: We will suppose that = ℵ and that there exists an ℵ -Lusin family in ℕ∗. Under these hypotheses, let be the Stone dual compact space of the Cohen-Parovičenko Boolean algebra. The definition of that

Boolean algebra implies that is an -space and thus () is 1- separably injective by Theorem (3.1.8). We show that it is not universally 1-separably injective. The argument follows the scheme, where they prove that does not map onto ℕ, but we use ℵ-Lusin families instead of -chains because they fit better in the functional analytic. Let {: ∈ ℕ} be a sequence of pairwise disjoint clopen subsets of , and let = ⋃ . Let ⊂ ℓ be the Banach space of convergent sequences, and : → () be the operator given by ()() if ∈ and ()() = lim if ≠ . → If () were universally 1-separably injective, we should have an extension : ℓ → () of with ‖‖ = 1 . We shall derive a contradiction from the existence of such operator. Notice that the conditions of Lemma (3.2.10) are applied, so is positive (observe that = (ℕ ∪ {∞} ) and = ∘ where : → ℕ ∪

{∞} is given by () = if ∈ and () = ∞ if ∉ ). For every ⊂ ℕ we will denote [] = ℕ\ℕ. The clopen subsets of ℕ∗ are exactly the sets of the form [], and we have that [] = [] if and only if (\) ∪ (\) is finite. ∗ Let ℱ be an ℵ-Lusin family in ℕ . For = [] ∈ ℱ and 0 < < , let = { ∈ \: (1)() > 1 − }. Let us remark that depends only on and not on the choice of . This is because if [] = [], then 1 − 1 ∈ , hence (1 − 1) = (1 − 1) which vanishes out of , so (1)|\ = 1|\.

Claim 1. If < and ∈ ℱ, then ⊂ . Claim 2. ∩ = for every ≠ . Proof of Claim 2. Since ∩ = ∅ we can choose , ⊂ ℕ such that = [], = [] and ∩ = ∅ . If ∈ ∩ , (1 + 1)() > 2 − 2 > 1 which is a contradiction because 1 + 1 = 1∪ and ‖(1∪)‖ ≤ ‖‖‖(1∪)‖ = 1. End of the Proof of Claim 2. For every ∈ ℱ, let be a clopen subset of \ such that . ⊂ ⊂ .. By the preceding claims, this is a disjoint family of clopen sets. It follows that \ does not contain any ℵ-Lusin family. Therefore we can find , ℋ ⊂ ℱ with || = |ℋ| = ℵ such that {̅: ∈ } ∩ {: ∈ ℋ} = .

Now, for every ∈ ℕ choose a point ∈ . Let : ℕ → be a continuous function such that () = . 1, if ∈ []; Claim 3. For ∈ ℕ, ⊂ ℕ, (1 )() = 0, if ∉ []. Proof of Claim 3. It is enough to check it for = ∈ ℕ. This is a consequence of the fact that is positive, because if ∈ , ∉ , then

0 ≤ (1) ≤ (1) ≤ 1ℕ{} ≤ 1. End of the Proof of Claim 3. The function g is one-to-one because {: ∈ } ∩ {: ∉ } = for every ⊂ ℕ, as the function (1) separates these sets. On the other hand, as a consequence of Claim 3 above, for every ∈ ℱ and every , () = , and also = . These facts make the families ℋ ∗ and above to contradict that ℱ is an ℵ-Lusin family in ℕ .

Chapter 4 Complex Vector Lattices

We motivates the definition of Complexification of Archimedean vector lattices, the Fremlin tensor product of Archimedean complex- vector lattices, and a theory of powers of Archimedean complex vector lattices.

Section (4.1): Vector Lattices Complexifications

A vector space () equipped with a partial order “≤” is called a vector lattices if for each pair , in : (i) There is smallest element (denoted by ∧ ) for which ≤ and ≤ , (ii) There is a largest element (dented by ∧ ) for which ≤ and ≤ , (iii) if ≤ then + ≤ + for all ∈ , (iv) If ≤ and ∈ ℝ then ≤ , the element || ≔ ∨ (−) is called the Modulus of . The element ≔ ∨ 0 is called the positive part of . The ≔ (−) ∨ 0 is called the negative part of . If a vector lattice is equipped with a norm ‖⋅‖ for which, (v) for all , ∈ , if || ≤ || then ‖‖ ≤ ‖‖ then (equipped with ≤ and ‖⋅‖) is called a normed vector lattice [8].

We discuss the specific case that we will use to complexify Archimedean vector lattices over ℝ and multilinear maps over ℝ. Using

|||| the notation, let (, ) = (, ∈ ℝ) . If is an ,

Archimedean vector lattice over ℝ and , ∈ then ,(, ) = ( ⊞ ) , where ⊞ : = sup{ cos + sin : ∈ [0, 2]} , as √ defined. Therefore, from Mittelmeyer and Wolff’s Theorem, a vector space + over ℂ is an Archimedean vector lattice over ℂ if and only if is a ,-complete Archimedean vector lattice over ℝ. We refer to the , , -completion ( , ) of as the square mean completion of .

Noting that , is absolutely invariant. We summarize the newly found information regarding functional completions for this special case in the following corollary of. Corollary (4.1.1) [4] If is an Archimedean vector lattice over ℝ then there exists a unique square mean completion (, , ) of . Moreover, if , … , , are Archimedean vector lattices over ℝ with square mean , completions , ( ∈ {1, … , }) and is square mean complete,

then for every vector lattice -morphism :× → there exists a , , unique vector lattice -morphism :× → such that , (), … , () = (, … , ). Furthermore, if is uniformly complete and :× → is a positive -linear map then there exists , , a unique positive -linear map :× → such that , (), … , () = (, … , ) for every ∈ ( ∈ {1, … , }). Here is the natural embedding of into . We now turn to complexifications of Archimedean vector lattices over ℝ. Definition (4.1.2) [4] For an Archimedean vector lattice over ℝ we define a pair |ℂ|, to be a vector lattice complexification of if the following hold.

(i) |ℂ| is an Archimedean vector lattice over ℂ. (ii) : → is an injective vector lattice ℝ-homomorphism. |ℂ| (iii) For every Archimedean vector lattice Fover ℂ as well as for

every vector lattice ℝ- homomorphism : → , there exists a unique vector lattice ℂ-homomorphism

|ℂ|: |ℂ| → such that |ℂ| ∘ = . We next prove the existence and uniqueness of vector lattice complexifications. Theorem (4.1.3) [4] If is an Archimedean vector lattice over ℝ then there exists a vector lattice complexification of , unique up to vector lattice isomorphism. Proof: Let be an Archimedean vector lattice over ℝ. By Corollary (4.1.1), there exists a unique square mean completion (, , ) of . , Define |ℂ| ≔ ( )ℂ and observe that |ℂ| is an Archimedean vector lattice over ℂ and that = , . Next, let be an Archimedean |ℂ| vector lattice over ℂ and let : → be a vector lattice ℝ - homomorphism. Since is square mean complete, there exists a unique , , , vector lattice ℝ-homomorphism : → such that ∘ = . , , Define |ℂ|: |ℂ| → by |ℂ|( + ) = () + () for every

+ ∈ |ℂ|. Then |ℂ| ∘ = . Moreover, for + ∈ |ℂ| we have from (see [4]) that

, , , |ℂ|(| + |) = ( ⊞ ) = () ⊞ () = |ℂ|( + ) .

Thus |ℂ| is a vector lattice C-homomorphism and therefore

|ℂ|, is a vector lattice complexification of . Next, we prove the uniqueness. To this end, suppose |ℂ|, and |ℂ|, are vector lattice complexifications of . Then , and , are |ℂ| |ℂ| square mean completions of , and hence there exists a vector lattice isomorphism : → . Similar to above, the map |ℂ| |ℂ| |ℂ|

ℂ: |ℂ| → |ℂ| defined by ℂ( + ) = () + () is a vector lattice ℂ-homomorphism. The bijectivity of ℂ is evident. For the square mean completion (,, ) of , we will from now on identify with () . Using this identification, we complexify positive ℝ -linear maps (respectively, vector lattice ℝ -morphisms) to positive ℂ -linear maps (respectively, vector lattice ℂ -morphisms) as follows. Let , … , , be Archimedean vector lattices over ℝ with square mean complete, and let :× → be a vector lattice ℝ - morphism. For ( + , … , + ) ∈× |ℂ|, define |ℂ|:× |ℂ| → ℂ by

∑ ( ) , |ℂ| + , … , + ≔ , … , . ∈{,} If is uniformly complete and above is any positive ℝ-linear map, we define |ℂ| in a similar manner. We collect a few facts regarding this complexification in the following proposition. Statement (iii) and the , statement that |ℂ| = ( )|ℂ| in (i) and (ii) are evident. The proof of (ii) follows from Corollary (4.1.1), and the proof of (i) is similar to the complexification of vector lattice homo-morphisms seen in the proof of Theorem (4.1.3).

Proposition (4.1.4) [4] Let , … , , be Archimedean vector lattices over ℝ with square mean complete. (i) If a map :× → is a vector lattice ℝ-morphism then |ℂ| , is a vector lattice ℂ-morphism and |ℂ| = ( )|ℂ|. (ii) If is uniformly complete and :× → is a positive ℝ - , linear map then |ℂ|is a positive ℂ-linear map and |ℂ| = ( )|ℂ|.

(iii) If in (i) or (ii) all , … , are square mean complete then |ℂ| = ℂ.

Section (4.2): The Archimedean Vector Lattice Tensor Product We define the tensor product of Archimedean vector lattices over and show the existence of the Archimedean complex tensor product by complexifying the Fremlin tensor product of Archimedean real vector lattices. We start with the definition of these tensor products of Archimedean vector lattices over . For = 2 and = ℝ the definition coincides with Fremlin’s definition of the Archimedean tensor product of Archimedean vector lattices over ℝ.

Definition (4.2.1) [4] Given Archimedean vector lattices , … , over , we define a pair ⊗ ,⊗ to be an Archimedean vector lattice tensor product of , … , if the following hold. (i) ⊗ is an Archimedean vector lattice over . (ii) ⊗ is a vector lattice -morphism. (iii) For every Archimedean vector lattice over and for every vector lattice - morphism :× → , there exists a uniquely ⊗ determined vector lattice homomorphism :⊗ → such that ⊗ ∘⊗= . Below and throughout the rest of this section, (⊗ ,⊗) denotes the algebraic tensor product of vector spaces , … , over . Lemma (4.2.2) [4] Let , … , be Archimedean vector lattices over ℝ. (i) There exists an essentially unique Archimedean vector lattice ⊗ over ℝ and a vector lattice -morphism ⊗:× → ⊗ such that for every Archimedean vector lattice over ℝ and every vector lattice -morphism :× → , there exists a ⊗ unique vector lattice homomorphism :⊗ → such that ⊗ ∘⊗= . (ii) There exists an injective linear map :⊗ →⊗ such that ∘⊗=⊗. (iii) For every ∈⊗ , there exist ∈ ( ∈ {1 … , }) such that for every > 0 , there exists ∈⊗ such that | − | ≤ ( ⊗ … ⊗ ), i.e. ⊗ is relatively uniformly dense in ⊗ .

(iv) For every 0 < ∈⊗ there exist ∈ ( ∈ {1 … , }) such that 0 < ( ⊗ … ⊗ ) ≤ , i.e. ⊗ is order dense in ⊗ . We deal with the existence and uniqueness of the complex Archimedean vector lattice tensor product and requires several prerequisite results. The next lemma surely is known. Lemma (4.2.3) [4] If , … , are vector spaces over ℝ then ⊗ ℂ and (⊗ )ℂ are isomorphic as vector spaces over ℂ. Proof: Since the algebraic tensor product is associative, we only need to prove the result for = 2, and use induction. The case = 2 is the content of Theorem (see [4]), but, we provide a sketch of van Zyl’s proof to correct some potential confusion caused by an accumulation of minor misprints. First let and be vector spaces over ℝ, and let ( ⊗ ,⊗) and (ℂ ⊗ ℂ) be the algebraic tensor products of , , respectively ℂℂ . Since ⊗ℂ: ℂ × ℂ → ( ⊗ )ℂ is a bilinear map over ℂ , it induces a unique ℂ-linear map : ℂ ⊗ ℂ → ( ⊗ )ℂ. It is easy to see that is surjective. To show that is injective, let = ∑( + ) ⊗ ( + ) ∈ ℂ ⊗ ℂ and suppose that () = 0. Note that () = ∑( ⊗ − ⊗ + ⊗ + ⊗ ), and so for any ℝ-linear functionals on and on we have ()() − ()() = 0 and ()() + ()() = 0 (∗) Let = + be a ℂ-linear functional on ℂ and let = + be a ℂ -linear functional on ℂ, both written in their natural decompositions. Then , are ℝ -linear functional on and , are ℝ -linear functionals on . Now

( + )( + ) = ()() − ()() − ()() + ()() + ()() + ()() + ()() − ()() − ()() + ()() − ()() − ()() + ()() − ()() − ()() + ()(). Applying (∗) again to each of these eight summands, we have that ∑ ( + )( + ) = 0. Therefore = 0 and is injective. Then is a vector space isomorphism. In light of the previous lemma, we will from now identify ⊗ with ⊗ for vector spaces , … , over ℝ. ℂ There exists a simpler construction of the square mean completion than the construction preceding Proposition (see [4]), which was given in a more general setting. Indeed, Azouzi constructs a square mean completion of an Archimedean vector lattice over ℝ essentially as follows. Let : = and for every ∈ ℕ , define : = ∪

,(, ): , ∈ , where ,(, ): , ∈ denotes the vector subspace of generated by ,(, ): , ∈ . Then define ⊞ ⊞ : = ⋃∈ℕ . To see that is a vector lattice, note that for every ∈

⊞ there exists ∈ ℕ such that ∈ . Then || = √2,(, 0) ∈ ⊞ ⊞ . It follows that is the square mean completion of , that is, and , are isomorphic as vector lattices. In fact, from the identity ,(, ) = ,(, ) for every ∈ ℝ and every , ∈ , we have = ∑ ,(, ): , ∈ . We use this fact in the first of the two following lemmas that are needed for Proposition (4.2.6). Lemma (4.2.4) [4] Denote the standard sine and cosine functions on 0, by sin and cos, respectively. For an Archimedean vector lattice , over ℝ and for every ∈ ( ) there exists , … , ∈ and { } ( { }) ,, … , , ∈ cos, sin ∈ 1, … , such that

= sup ,() . ∈, Proof: Our proof is via mathematical induction. Let ℎ ∈ and first suppose that = ,(, ) for some , ∈ . Then = sup{ cos + sin : ∈ [0, /2]}. Next, suppose that = ∑ ,(, ). Then

= sup { cos + sin } ∈,

= sup ( cos + sin ) . . ∈, This completes the base step of the induction argument. For the inductive step, suppose that for every ∈ there exists , … , ∈ and { }( { }) , … , ∈ cos, sin ∈ 1, … , such that

= sup ,, . ∈, , Let ∈ . From the argument in the base step above, we may assume that = ,(, ) for some , ∈ . By the induction hypothesis there exists , … , , , … , ∈ and { }( { }) ,, … , , , ,, … , , ∈ cos, sin ∈ 1, … , such that

= sup ,, and ,∈ ,

= sup ,,. ∈, , Then

= , sup ,, , sup ,, ∈, ∈, , ,

= sup sup ,, cos ∈, ∈, ,

+ sup ,, sin ∈, ,

= sup ,, cos + ,, sin . , ∈, , The next lemma can be verified using mathematical induction. We do not include the Proof:

Lemma (4.2.5) [4] Let , … , be Lipschitz functions on ℝ with Lipschitz constant 1. Also assume that |()| ≤ 1 for every ∈ {1, … , } and every ∈ ℝ. Then for every , ∈ ℝ( ∈ {1, … , }) we have |∏ ( ) − ∏ ()| ≤ ∑| − |. We have the following proposition. Proposition (4.2.6) [4] If is an Archimedean vector lattice over ℝ then is relatively uniformly dense in , . Proof: Let be an Archimedean vector lattice over ℝ and first suppose

, that ∈ ( ) . Say that = sup ∑ ∏ ,, for ∈, , { }( { }) some , … , ∈ and ,, … , , ∈ cos, sin ∈ 1, … , . Note that given ∈ 0, ∈ ℕ there exist ∈ ℕ such that , − , , ≤ . Since sine and cosine are both Lipschitz functions with , Lipschitz constant 1 we have from Lemma (4.2.5) that

, − , , , 2

, ≤ − | | , , , 2 , ≤ − | | ≤ | |. , 2 2 Thus, , ≤ + | | , , , 2 2 , ≤ + | |. , 2 2 , Since this is true for every ∈ 0, ( ∈ {1, , … , }, ∈ {1, … , }) we , have , 0 ≤ − ≤ | | . 2 2 , It follows that the sequence : = ⋁ ∑ ∏ , converges , relatively uniformly to .

Finally, for ∈ , there exist sequences (), () in such that and . Then − . We are ready to deal with the Archimedean tensor product of Archimedean vector lattices over .

Theorem (4.2.7) [4] Let , … , be Archimedean vector lattices over . (i) There exists an essentially unique Archimedean vector lattice ⊗ over and a vector lattice -morphism ⊗:× → ⊗ such that for every Archimedean vector lattice over and every vector lattice -morphism :× → , there exists a ⊗ unique vector lattice homomorphism :⊗ → such that ⊗ ∘⊗= . (ii) There exists an injective linear map :⊗ →⊗ such that ∘⊗=⊗.

(iii) ⊗ ,⊗ ≤ 2 . Thus, ⊗ is dense in ⊗ in the relatively uniform topology. (iv) For every ∈ ⊗ \{0}, there exist ⊗ … ⊗ ∈ ( { }) ( ⊗ with ∈ ∈ 1 … , such that 0 < ⊗ … ⊗ ) ≤ ||, i.e. ⊗ is order dense in ⊗ . Proof: By Lemma (4.2.2), statements (i)-(iv) are valid for = ℝ. We assume in the proof below that = ℂ.

(i) Let , … , , be Archimedean vector lattices over ℂ. Denote by ⊗ ,⊗ the Archimedean vector lattice tensor

product of , … , . We claim that the pair ⊗ ,⊗|ℂ| is the unique Archimedean complex |ℂ| vector lattice tensor product of , … , . Let :× → be a vector lattice -morphism. From Lemma (4.2.2), the map ⊗ ⊗ induces a unique vector lattice homomorphism on ⊗ ⊗ ⊗ such that ∘ ⊗= . Also, the map extends ⊗ , uniquely to a vector lattice homomorphism on , ⊗ (Corollary (4.1.1)). By Proposition (4.1.4) (i), ⊗ the map ⊗|ℂ| is a vector lattice - morphism and is a |ℂ| vector lattice homomorphism. We will prove that the map ⊗ : ⊗ → is the unique vector lattice |ℂ| |ℂ| ⊗ homomorphism such that ∘ ⊗|ℂ|= . Indeed, for every |ℂ| ( + , … , , ) ∈× we have ⊗ ∘ ⊗|ℂ| ( + , … , , ) |ℂ|

⊗ ∑ = ⊗ , … , |ℂ| ∈{,} ⊗ ∑ = ∘ ⊗ , … , { } ∈ , ∑ ( ) = , … , = + , … , + . ∈{,}

Since every vector lattice ℂ-homomorphism is real, the uniqueness of ⊗ ⊗ , follows from the uniqueness of . |ℂ| The proof of uniqueness of the Archimedean complex vector lattice tensor product is not different from the real case. (ii) Consider the newly minted tensor product ⊗ ,⊗ constructed in (i). By Lemma (4.2.2), there exists an Archimedean vector lattice over ℝ and a vector lattice -morphism :× → such ⊗ that the induced linear map :⊗ → is injective. By taking the square mean completion of , if necessary, we will assume that is square mean complete. By taking vector space complexifications, we find ⊗ an injective vector lattice -morphism : ⊗ → ℂ , or ℂ ℂ equivalently by Lemma (4.2.3), ⊗ :⊗ → . Moreover, if ℂ ℂ ⊗ (ℂ) ∶ ⊗ → is the unique linear map induced by ℂ, then for + ∈ ( ∈ {1, … , }), ( )⊗( + ⊗ … ⊗ + ) = ( + , … , + ) ℂ ℂ ∑ = , … , ∈{,} ⊗ ∑ = ⊗ … ⊗ . ∈{,} In particular, ( )⊗ is a real map with ( )⊗ = ⊗, and therefore ℂ ℂ we have ( )⊗ = ⊗ . From part (i) of this theorem there exists a ℂ ℂ ⊗ unique vector lattice ℂ-homomorphism (ℂ) :⊗ → ℂ such that ⊗ (ℂ) ∘ ⊗= ℂ . Moreover, there exists a unique ℂ -linear map ⊗ :⊗ →⊗ such that ∘ ⊗=⊗. Then (ℂ) ∘ ∘ ⊗= ℂ , and hence ( )⊗ ∘ = ( )⊗ = ⊗ . Therefore is injective. ℂ ℂ ℂ (c) By Lemma (4.2.2) we know that ⊗ is relatively uniformly dense in ⊗ . We also know from Proposition (4.2.6) , that ⊗ is relatively uniformly dense in ⊗ . By taking vector space complexifications, we have ⊗ ,⊗ ≤ 2.

, { } | | (d) Suppose ∈ ⊗ \ 0 . Then 0 < ∈ ⊗ . Since ⊗ is order dense in ⊗ , it is also order dense in , ⊗ . Thus there exists ∈⊗ such that 0 < ≤ | | . From Lemma (4.2.2), there exists ⊗ … ⊗ ∈⊗ with ∈ ( ∈ {1, … , }) such that 0 < ( ⊗ … ⊗ ) ≤ . In (i) it is necessary to take the vector lattice complexification of ⊗ to ensure that ⊗ is an Archimedean vector lattice |ℂ| over ℂ. Indeed, Theorems (4.2.10) and (4.2.11) furnish examples where the vector space complexification ⊗ does not suffice. |ℂ| We need two lemmas first. Lemma (4.2.8) [4] Let and be nonempty subsets of ℝ without isolated points. Then the function : (, ) ⟼ + (, ) ∈ × is in the square mean completion of () ⊗ () but for all nonempty open subsets of and of we have |× ∉ () ⊗ (). Proof: For ∈ () and ∈ () we identify ⊗ with the function (, ) ↦ ()()(, ) ∈ × . Consider the element of the square mean completion of () ⊗ () defined by (, ) ⟼ + (, ) ∈ × . Let and be open nonempty subsets of and , respectively. We will show that the vector subspace of () generated by {(·, ): ∈ }, whose elements are considered as functions on , is not finite- dimensional. It follows that |× ∉ () ⊗ (). Since is open and nonempty and has no isolated points, we can choose ∈ (for all ∈ ℕ) for which ≠ when ≠ . Let ∈ ℕ and let , … , ∈ ℝ for which + = (, ) = 0 for all ∈ . Since the function ⟼ + ( ∈ ℝ) is times differentiable at every ∈ , a routine calculation shows that the × matrix () defined by () = when evaluated at the vector (, … , ) yields the vector (0, … ,0) for every non-zero ∈ . However,

∏ + det() = det()where the × matrix ( ) is defined by () = , which has (Vandermonde) determinant 1 1 − − = ≠ 0. + + + ( + ) Thus det() ≠ 0 for every non-zero ∈ , the vector subspace of () generated by {(·, ): ∈ } (as functions on ) is infinite dimensional, and |× ∉ () ⊗ (). Lemma (4.2.9) [4] Let and be nonempty subsets of ℝ without isolated points and let ∈ () ⊗ (). Then there exists a nonempty open subset of × and ∈ () ⊗ () such that | = |. Proof: Note that () ⊗ ()is the vector lattice generated by () ⊗ () in ( × ). Every element ∈ () ⊗ () is of the form = ⋀ ⋁ , where , ∈ () ⊗ () for each and each . Let

, ∈ () ⊗ (). If ≠ , we may assume there exists (, ) such that (, ) < (, ) and then there exists a nonempty open subset of × such that ∧ = on . Of course such an open subset also exists if = . By repeating this argument there exists a nonempty open set ⊆ such that ⋀⋁ , = ⋁ , on for some ∈ {1 … , }. Similarly, there exists a nonempty open set ⊂ such that ⋁ , = { } , on for some ∈ 1, … , . Theorem (4.2.10) [4] Let and be nonempty subsets of ℝ without isolated points. Then () ⊗ () is not square mean complete. Therefore () ⊗ () is not an Archimedean vector lattice over ℂ. ℂ Proof: Assume that the element of Lemma (4.2.8) is in () ⊗ (). Then by Lemma (4.2.9) there exists a nonempty open set in × and an element ∈ () ⊗ () such that | = |. However the open set contains a nonempty open subset of the form × with 0 ∉ . This contradicts Lemma (4.2.8). We use Theorem (4.2.10) to prove the following. Theorem (4.2.11) [4] If and are uncountable compact metrizable spaces then () ⊗ () is not square mean complete. Therefore () ⊗ () is not an Archimedean vector lattice over ℂ. ℂ

Proof: By [4], we know that both and contain a closed subset homeomorphic with the Cantor set . Then × can be viewed as a closed subset of × and the function : (, ) ⟼ + (, ) ∈ × is continuous. By Tietze’s Extension Theorem, the function ⟼ ( ∈ ) can be extended to continuous functions and on and , respectively. Then the function : (, ) ⟼ () + () (, ) ∈ × is a continuous function in the square mean completion of

() ⊗ () that extends . If were in () ⊗ () itself then its restriction to × would be in () ⊗ () which by Theorem (4.2.10) is impossible. This proves the theorem. It is certainly tempting to conjecture the following. Conjecture (4.2.12) [4] If and are infinite compact metrizable spaces then () ⊗ () is not square mean complete. The above two theorems show that the old way of complexifying Archimedean vector lattices via vector space complexifications is inadequate for pursuing complex analysis on Archimedean complex vector lattices. We remark that the complex Archimedean vector lattice tensor product, like its real counterpart, possesses as well a universal property with respect to positive multilinear maps and complex uniformly complete vector lattices as range. The proof of this universal property, stated in the theorem below, is similar to the proof of Theorem (4.2.7) (i).

Theorem (4.2.13) [4] Let , … , , be Archimedean vector lattices over with uniformly complete. If :× → is a positive - linear map, then there exists a unique positive - linear map ⊗ ⊗ :⊗ → such that ∘⊗= . A reformulation of part (i) of Theorem (4.2.7) in terms of Archimedean real vector lattices and vector lattice complexifications is the following.s

Theorem (4.2.14) [4] Let , … , , be Archimedean vector lattices over ℝ and suppose that :× → is a vector lattice ℝ-morphism. ⊗ There exists a unique vector lattice ℂ-morphism |ℂ| :⊗ |ℂ| → ⊗ |ℂ| such that |ℂ| ∘ ⊗ = . × Proof: Consider to be a vector lattice ℝ-morphism from × to , . By Proposition (4.1.4) (i) there exists a unique vector lattice ℂ-

⊗ morphism |ℂ|:× |ℂ| → |ℂ| such that |ℂ| = . If |ℂ| is × ⊗ the unique vector lattice ℂ-homomorphism induced by ℂ then |ℂ| ∘ ⊗ ⊗= ℂ. In particular, |ℂ| ∘ ⊗ = . ×

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