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University of Alberta Operator Ideals on Ordered Banach Spaces University of Alberta Operator ideals on ordered Banach spaces by Eugeniu Spinu A thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics Department of Mathematical and Statistical Sciences c Eugeniu Spinu Spring, 2013 Edmonton, Alberta Permission is hereby granted to the University of Alberta Libraries to reproduce single copies of this thesis and to lend or sell such copies for private, scholarly or scientific research purposes only. Where the thesis is converted to, or otherwise made available in digital form, the University of Alberta will advise potential users of the thesis of these terms. The author reserves all other publication and other rights in association with the copyright in the thesis and, except as herein before provided, neither the thesis nor any substantial portion thereof may be printed or otherwise reproduced in any material form whatsoever without the author's prior written permission. To my family Abstract In this thesis we study operator ideals on ordered Banach spaces such as Banach lattices, C∗-algebras, and noncommutative function spaces. The first part of this work is concerned with the domination problem: the relationship between order and algebraic ideals of operators. Fremlin, Dodds and Wickstead described all Banach lattices on which every operator dominated by a compact operator is always compact. First, we show that even if the dominated operator is not compact it still belongs to a relatively small class of operators, namely, the ideal of inessential operators. A similar question is studied for strictly singular operators. In particular, we show that the cube of every operator, dominated by a strictly singular operator, is inessential. Then we provide a complete solution of the domination prob- lem for compact and weakly compact operators acting between C∗-algebras and noncommutative function spaces. Finally, we consider the domination problem for weakly compact operators acting on general noncommutative function spaces. The second part is devoted to the operator ideal structure of the alge- bra of all linear bounded operators on a Banach space. First, we investigate the existence of non-trivial proper ideals on Lorentz sequence spaces and characterize some of them. Second, we look at the coincidence of some clas- sical operator ideals, such as of compact, strictly singular, innesential, and Dunford-Pettis operators acting on noncommutative Lp-spaces. In particu- lar, we obtain a characterization of strictly singular and inessential operators acting either between discrete noncommutative Lp-spaces or Lp-spaces, as- sociated with a hyperfinite von Neumann algebras with finite trace. Many of the results presented in this thesis were obtained jointly with other people. The thesis is based on papers [58, 70, 71, 93] by the author and his collaborators. Acknowledgements First and foremost, I would like to express my deep gratitude to my adviser, Vladimir G. Troitsky, who guided me through the PhD program, broadened my mathematical horizons, exposed me to the mathematical community, and bravely fought my written English. He has been supportive far beyond mathematics. I would like to thank Timur Oikhberg, who has been more to me than just a collaborator, for introducing me to operator spaces and for the most stimulating correspondence on various mathematical topics. I thank Anna Kaminska, Alexey Popov, Adi Tcaciuc for the results we proved together that I included in my thesis. I also want to thank B. de Pagter for proofreading the thesis. Contents 1 Basic definitions and notations 1 1.1 Ordered Banach spaces . 1 1.2 Banach lattices . 2 1.3 C∗ and von Neumann algebras . 3 1.4 Noncommutative function spaces . 6 1.5 Operator ideals . 8 2 Domination problem 10 2.1 Introduction . 10 2.2 Banach lattices . 12 2.3 Domination problem in the noncommutative setting . 20 2.3.1 Compactness and positivity in Schatten spaces . 21 2.3.2 Compactness of order intervals in C∗-algebras . 26 2.3.3 Positive Schur Property. Compactness of order inter- vals in Schatten spaces . 31 2.3.4 Compactness of order intervals in preduals of von Neu- mann algebras . 36 2.3.5 Compact operators on noncommutative function spaces 41 2.3.6 Compact operators on C∗-algebras and their duals . 44 2.3.7 Comparisons with multiplication operators . 48 2.3.8 Weakly compact operators . 53 3 Operator ideals 57 3.1 Operator ideals on Lorentz sequence spaces . 58 3.1.1 Operators factorable through `p . 63 3.1.2 Strictly singular operators . 65 3.1.3 Operators factorable through the formal identity . 70 3.1.4 dw;p-strictly singular operators . 80 3.1.5 Strictly singular operators between `p and dw;p. 84 3.2 Strictly singular operators on noncommutative Lp . 93 3.2.1 Noncommutative Lp: continuous case . 94 3.2.2 Noncommutative Lp: discrete case . 106 3.2.3 Operator ideals on C∗-algebras and function spaces . 114 4 Summary 120 Chapter 1 Basic definitions and notations 1.1 Ordered Banach spaces Let X be a Banach space and C ⊂ X be a cone, that is, a set closed un- der addition and multiplication by a non-negative real number such that C \ (−C) = f0g. This cone is generating in X if X = C − C. The set of all ∗ ∗ functionals f 2 X , such that f(x) > 0 for every x 2 C forms a cone in X , that is referred as the dual cone of C. If the dual cone of C is generating then we call C normal. We say that X is a real ordered Banach space (OBS) if it is equipped with a norm-closed cone C ⊂ X. The elements of C are called positive (denoted x > 0). One can define an order as follows: x 6 y if and only if y−x 2 C. An order interval [x; y] ⊂ X is the set of all elements z 2 X such that x 6 z 6 y. It was shown in [8] that a closed cone C is generating if and only if there exists δX > 0, such that, for every x 2 X, there exist a and b such that x = a − b and max(kak; kbk) 6 δX ka − bk for every a; b 2 C. And it is normal if and only if there exists γX > 0 such that kxk 6 γX maxfkak; kbkg, whenever a 6 x 6 b. 1 An operator T acting between two OBS's is called positive if it maps positive elements to positive elements. A complex OBS Y is the complexification of a real OBS Yr that is Y = Yr + iYr. The positive elements of Y are exactly those belonging to the cone of Yr. We say a cone is generating (normal) in Y if it is generating (normal) in Yr. Natural examples of OBS with proper normal generating cones and δ = γ = 1 are Banach lattices, C∗-algebras, and noncommutative functions spaces. They will be discussed in the following sections of this chapter. 1.2 Banach lattices We adhere to standard definitions and properties of Banach lattices that can be found in [1, 6, 63, 64, 67] An OBS E is called a (real) Banach lattice with respect to its cone if for every two elements x; y 2 E • their supremum x _ y and infimum x ^ y are in E, • jxj 6 jyj implies kxk 6 kyk, where jxj = x _ (−x). For a Banach lattice E its positive cone E+ = fx 2 E : x > 0g is normal and generating. A Banach lattice E is called order continuous if kxαk ! 0 for every decreasing net (xα) ⊂ E such that ^xα = 0. A linear subspace I ⊆ E is an (order) ideal if x 2 I whenever y 2 E and jxj < jyj. A non-zero element a 2 E+ is an atom if 0 6 b 6 a implies b = λa, 2 for some constant λ > 0. We say E is atomic if for every element x 2 E+, there exists an atom a 6 x in E. Any Banach space with a 1-unconditional basis can be realized as an order continuous atomic Banach lattice with the standard order defined co- ordinatwise. There are many examples of non-atomic order continuous Ba- nach lattices, in particular, Lp (1 6 p < 1), and various Lorentz and Orlicz function spaces on non-atomic measure space. An element e 2 E is called an (order) unit if any ideal that contains e coincides with E. We say that E is an AM-space if kx+yk = maxfkxk; kykg for every disjoint x; y 2 E. By Kakutani's theorem every AM- space with a unit is lattice isometric(there exists a surjective isometry that preserves lattice operations) to a C(K) space for some compact Hausdorff topological space K. 1.3 C∗ and von Neumann algebras A C∗-algebra A is a Banach algebra with a ∗-map from A to A with the following properties: (i) (A + B)∗ = A∗ + B∗, (ii) (cA)∗ =cA ¯ ∗, (iii) A∗∗ = A, (iv) (AB)∗ = B∗A∗, (v) kA∗Ak = kAk2, 3 for every A; B 2 A and c 2 C. A C∗-algebra A is called unital if it has an algebraic unit 1. Every C∗- algebra A can be realized as a C∗-subalgebra of codimension one of a unital ∗ C algebra AU . This process is called the unitization. Thus, we can define the spectrum of A 2 A, σ(A) = fλ 2 C : λ1 − A is not invertible in AU g We say that A 2 A is positive if and only if it is self-adjoint (A = A∗) and the σ(A) ⊂ [0; 1).
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