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General Reflexivity for Absolutely Convex Sets University of New Hampshire University of New Hampshire Scholars' Repository Doctoral Dissertations Student Scholarship Spring 2019 General Reflexivity orF Absolutely Convex Sets Mahtab Lak University of New Hampshire, Durham Follow this and additional works at: https://scholars.unh.edu/dissertation Recommended Citation Lak, Mahtab, "General Reflexivity For Absolutely Convex Sets" (2019). Doctoral Dissertations. 2454. https://scholars.unh.edu/dissertation/2454 This Dissertation is brought to you for free and open access by the Student Scholarship at University of New Hampshire Scholars' Repository. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of University of New Hampshire Scholars' Repository. For more information, please contact [email protected]. GENERAL REFLEXIVITY FOR ABSOLUTELY CONVEX SETS BY MAHTAB LAK BS, Applied Mathematics, Sheikh-Bahaee University of Isfahan, Iran, September 2008 MS, Mathematics, Isfahan University of Technology, Iran , 2012 DISSERTATION Submitted to the University of New Hampshire in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Mathematics May 2019 ALL RIGHTS RESERVED ©2019 Mahtab Lak This dissertation has been examined and approved in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics by: Dissertation Director, Don Hadwin, Professor of Mathematics and Statistics Eric Nordgren, Professor of Mathematics and Statistics Junhoa Shen, Professor of Mathematics and Statistics John Gibson, Associate Professor of Mathematics and Statistics Mehment Orhan, Senior Lecture of Mathematics and Statistics on 4/15/2019. Original approval signatures are on file with the University of New Hampshire Graduate School. iii ACKNOWLEDGMENTS Firstly, I would like to express my sincere gratitude to my advisor Prof. Don Hadwin for the continuous support of my Ph.D study and related research, for his patience, motivation, and immense knowledge. His guidance helped me in all the time of research and writing of this thesis. I could not have imagined having a better advisor and mentor for my Ph.D study. Besides my advisor, I would like to thank the rest of my thesis committee: Prof. Eric Nordgren, Prof. Junhoa Shen, Prof. John Gibson and Prof. Mehmert Orhan. I wish to express my warmest thanks to all friends who made me enjoy pleasant and comfort- able life at UNH, including, but not restricted to: Alex, Eric, Gabi, Wenjing, Haihui, Prof. H, Jan, Jennifer. Finally, I would like to thank my family: Nathaniel Lanning and my mother and my father for supporting me spiritually throughout writing this thesis and my my life in general. iv TABLE OF CONTENTS Page ACKNOWLEDGMENTS ......................................................... iv ABSTRACT ..................................................................... vi CHAPTER 1. INTRODUCTION ............................................................ 1 2. AC-E-REFLEXIVITY ......................................................... 3 2.1 Reflexivity Triples.........................................................3 2.2 Definition of ac-E-reflexivity...............................................4 2.3 Hereditary ac-E-reflexivity.................................................8 2.4 Relative ac-E-reflexivity.................................................. 11 2.5 Reflexive Graphs......................................................... 12 3. AC-E-HYPERREFLEXIVITY ................................................ 15 3.1 Relative ac-E-hyperreflexivity.............................................. 26 3.2 Hereditary Hyperreflexivity................................................ 28 3.3 A New Result............................................................ 30 4. DIRECT INTEGRALS ....................................................... 37 5. APPLICATIONS............................................................. 52 5.1 Operators on Dual Pairs................................................... 52 5.2 Approximate Reflexivity.................................................. 54 BIBLIOGRAPHY ............................................................... 60 v ABSTRACT GENERAL REFLEXIVITY FOR ABSOLUTELY CONVEX SETS by Mahtab Lak University of New Hampshire, May, 2019 In 1994 D. Hadwin introduced the notion of a reflexivity triple (X; Y; E) and defined a very general notion of reflexivity, called E-reflexivity, for a linear subspace of the vector space X. Had- win’s general version included many special versions (algebraic reflexivity, topological reflexivity, approximate reflexivity, hyperreflexivity) that had been studied for linear spaces of linear transfor- mations on a vector space or Hilbert space. In this thesis we extend Hadwin’s notion and define and study the abstract notion of reflexivity for an absolutely convex subset of X. We have extended many of Hadwin’s results and obtained some new ones. We have also extended Hadwin’s gener- alized notion of direct integrals from measurable families of linear spaces to absolutely convex sets. vi CHAPTER 1 INTRODUCTION In [18] Paul Halmos introduced that notion of a reflexive algebra of operators on some Hilbert space based on the functor AlgLat. Since that time there was a flurry of research done on reflexive algebras and operators (an operator was called reflexive if the weak operator closed unital algebra it generates is reflexive). Later, and algebraic notion of reflexivity was initiated by D. Hadwin [13], although earlier examples had been discussed ([9],[10]) . R. L. Moore and K. J. Harrison introduced an asymptotic version of AlgLat and posed an asymptotic version of the reductive opertor problem. This problem was solved by C. Apostol, C. Foia¸sand D. Voiculescu [2], [3]. Later D. Hadwin investigated this asymptotic version of AlgLat, called ApprAlgLat and proved a number of results, including an asymptotic version of von Neumann’s famous double commutant theorem. In 2002 B. Prunaru [22] proved an asymptotic version of the reductive algebra problem. The notion of reflexivity was defined and studied for linear spaces of operators, and a lot of research was done on this topic. In [5] the notion of hyperreflexivity was introduced for linear spaces of operators on a Hilbert space (See also [20]), which was motivated by W. Arveson’s wonderful "distance formula" for nest algebras [4]. The question of whether every von Neumann algebra on a Hilbert space is hyper- reflexive turns out to be equivalent to D. Kadison’s famous "similarity problem" (is every bounded unital homomorphism from a C*-algebra into some B (H) similar to a ∗-homomorphism). In [5] W. Arveson proved that an ultraweakly closed linear subspace S of B (H) is hyperreflexive if and only if every ultraweakly continuous linear functional that annihilates S is an `1-sum of a sequence of rank-one tensors that annihilate S. Also D. Hadwin [15] introduced an analogue of hyperreflexivity for approximate reflexivity and proved that every unital C*-subalgebra of B (H) is approximately hyperreflexive. 1 In 1994 [14] Hadwin introduced the notion of a reflexivity triple (X; Y; E) and developed a general framework in which all of the above notions of reflexivity are special cases. In this paper Hadwin proved many theorems in this special setting that were known for B (H), were now ex- tended to many situations. Often the proofs in the general setting were simpler or, at least no more difficult, than those in the Hilbert space setting. Hadwin also introduced a notion of hyperreflexiv- ity theorem and generalized many know theorems, including a version Arveson charactization in terms of `1-sums of rank-one tensors whose proof was an application of the Hahn Banach theorem and did not rely on the spectral theorem for compact selfadjoint operators. Hadwin [14] also de- veloped a very general notion of direct integrals and proved some results that were new even in the Hilbert space setting. E. Azoff and H. A. Sheheda [7], [8] and Sheheda [28] studied a notion of reflexivity for convex sets of operators. In this thesis we address reflexivity and hyperreflexivity for absolutely convex sets in the framework of reflexivity triples. We also develop a notion of direct integrals for convex sets. We have extended many of the results in [14] and we prove some results that are new even for linear spaces. In section 2.1 we introduce Hadwin’s notion of a reflexity triple and the corresponding notion of reflexivity. Since we need to discuss convexity, the filed involved must contain [0; 1], so we restrict to the case where the field is either R or C. In section 2.2 we define the notion of ac-E-reflexivity for an absolutely convex set. 2 CHAPTER 2 AC-E-REFLEXIVITY 2.1 Reflexivity Triples Suppose that F is a topological field with a Hausdorff topology, e.g., F could be a subfield of the complex numbers with the usual topology, or F could be an arbitrary field with the discrete topology. Next suppose that X is a vector space over F and Y is a vector space of linear maps from X to F that sepreates the points of X (i.e., \ fker f : f 2 Y g = f0g). We call such a pair (X; Y ) a dual pair over F. We define the σ(X; Y )-topology on X to be the smallest topology on X that makes all ofthe maps in Y continuous on X. Since we can view X as a vector space of ( evaluation) functional on Y , we can define the σ(Y; X)-topology on Y to be the smallest topology for which the map f 7! f(x) is contimuous on Y for each x in X: If A ⊂ X and B ⊂ Y , we can define ? A = ff 2 Y : fjA = 0g and B? = \ fker f : f 2 Bg . If D is a subset of either X or Y , then sp(D) denotes the linear span of D, and sp(D) denotes the closure of sp(D) in either the σ(X; Y ) or σ(Y; X) topology. The following familiar results from the duality theory of real or complex topological vector spaces hold in this more general setting. Proposition 1. (1) A linear map f : X ! F is σ(X; Y )-continuous if and only if ker(f) is σ(X; Y )-closed. 3 (2) A linear map from X to F is σ(X; Y )-continuous if and only if it is in Y: (3) If M is a linear subspace of X and g : M ! F is linear and σ(X; Y )-continuous, then there is an f in Y such that fjM = g. ? (4) If A ⊂ X, then sp(A) = A ?. ? (5) If B ⊂ Y , then sp(B) = (B?) .
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