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COMMUTING MAPS on SOME SUBSETS THAT ARE NOT CLOSED UNDER ADDITION a Dissertation Submitted to Kent State University in Partial F COMMUTING MAPS ON SOME SUBSETS THAT ARE NOT CLOSED UNDER ADDITION A dissertation submitted to Kent State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy by Willian V. Franca August 2013 Dissertation written by Willian V. Franca B.S., Universidade Federal do Rio de Janeiro Janeiro (UFRJ), Rio de Janeiro, Brazil, 2005 M.S., Universidade Federal do Rio de Janeiro Janeiro (UFRJ), Rio de Janeiro, Brazil, 2008 Ph.D., Kent State University, 2013 Approved by |||||||||||||| Chair, Doctoral Dissertation Committee Dr. Mikhail Chebotar |||||||||||||| Member, Doctoral Dissertation Committee Dr. Richard Aron |||||||||||||| Member, Doctoral Dissertation Committee Dr. Artem Zvavitch |||||||||||||| Member, Outside Discipline Dr. Sergey Anokhin |||||||||||||| Member, Graduate Faculty Representative Dr. Arden Ruttan Accepted by |||||||||||||| Chair, Department of Mathematical Sciences Dr. Andrew Tonge |||||||||||||| Associate Dean, College of Arts and Sciences Dr. Raymond A. Craig ii TABLE OF CONTENTS ACKNOWLEDGEMENTS ..............................v INTRODUCTION ...................................1 1 Preliminaries .....................................3 1.1 Additive Commuting Maps . .3 1.2 Commuting Traces of Multiadditive Maps . .4 2 Additive Commuting Maps on The Set of Invertible or Singular Matrices 6 2.1 Additive Commuting Maps on The Set of Singular Matrices . .6 2.2 Additive Commuting Maps on The Set of Invertible Matrices . .7 3 Commuting Traces of Multiadditive Maps on Invertible or Singular Ma- trices .......................................... 11 3.1 Commuting Traces on The Set of Invertible Matrices . 11 3.2 Commuting Traces on The Set of Singular Matrices . 15 3.3 Applications . 16 3.4 Herstein's Conjecture . 18 4 Commuting Traces of Multiadditive Maps on The Set of Rank-k Matrices 20 4.1 Additive Commuting Maps on The Set of Rank-k Matrices . 20 4.2 Commuting Traces on The Set of Rank-k Matrices . 23 5 Commuting Traces on Invertible or Singular Operators .......... 27 5.1 Commuting Traces on Invertible Operators . 27 iii 5.2 Commuting Traces on Singular Operators . 31 BIBLIOGRAPHY . 33 iv ACKNOWLEDGEMENTS Thanks to my beloved wife for her love, understanding and support. Everything was just possible because she was by my side throughout my journey. She was there to brake my falls and to help me to get back on my feet after falling. To my brother Rodolfo for all the support during my undergrad studies. To my mom for being so brave rasing me and my brother by herself for so many years. I will never forget it. To my former step father, Almir for his financial support during my undergrad studies, without him I would not be able to get my bachelor degree. To my father, Brasilio who was passed away so long ago, wherever he is now I know that he is proud of the man that I have become. To Luiza, my Master degree advisor, for being so helpful during my masters. She helped in so many ways that it would take more than a thank note to explain. Thanks to her, Artem, and Richard everything worked out and I had the opportunity of studying here at the Kent State University. I want also to thank Misha for all his help with my research either suggesting interesting problems for me to work on or when he was just proofreading my papers, which I believe that was a very tedious work. I also want to thank him for answering my emails with all my questions and requests. I would terribly remiss not thanking him for his valuable fishing lessons that he taught to me during last summer. Some of the math learned during my Ph.D I will probably forget, but those lessons are unforgettable. v INTRODUCTION In 1961, during his AMS Hour talk, I. N. Herstein asked about the characterization of all additive maps from a unital simple ring to another one that preserve the multiplicative commutator. In particular, such maps preserve the set of invertible elements. In the context of matrix rings over fields and assuming that the map is linear we do have a Linear Preserver Problem. Thus, in the matrix ring setting, we can connect Herstein's conjecture and Linear Preserver Problems. Linear Preserver Problems is a very active area. This is true because the formulation of the problems is simple and natural, having quite often nice solutions. It is also common to refine known results by weakening the assumptions and to extend existing results to other matrix spaces or algebras. In [3], M. Breˇsarshowed how commuting maps also naturally appear when dealing with some Linear Preserver Problems, and there is no doubt that this is a relevant area of application for commuting maps. Our aim will be to use Breˇsar'stechnique to investigate Herstein's question in the case that our ring is the matrix ring over a field. In order to achieve our goal, we will start studying additive commuting maps on the set of invertible or singular elements. We will show that the only maps with such properties are the standard ones. Simultaneously, we will be demonstrating how results on functional identities [4] can be extended to sets that are not closed under addition. Later, we will describe multiadditive maps that are commuting on the set of invertible or singular elements. Such description combined with Breˇsar'sidea will lead us to draw some conclusions about Herstein's conjecture when R is the matrix ring over a field K (some assumptions about K are necessary). Finally, we will deal with commuting maps on the set of rank-k matrices. This is not 1 2 only interesting because we will generalize some of the results obtained before but also because in the case that k = 1 we will see that there are commuting maps on the set of the rank-1 matrices that are not of the standard form. The fifth and last chapter will be devoted to talk about commuting maps on the sets of invertible or singular elements when the ring R is the ring of all bounded operators from a real or complex separable Hilbert space H to itself. CHAPTER 1 Preliminaries In this work n will denote a natural number greater or equal than 2, and Mn(K) will represent the ring of all n × n matrices over a field K with center Z = K · I, where I is the identity matrix. Furthermore, it should be mentioned that the set of all nonzero elements ∗ of K will be denoted by K . 1.1 Additive Commuting Maps Definition 1. Let R be a ring. An additive map f : R ! R is called commuting if f(r)r = rf(r), equivalently, [f(r); r] = 0 for all r 2 R. There are some obvious examples of commuting maps, like additive maps µ : R ! C having range in the center C of R and maps of the form f(r) = λr + µ(r), where λ 2 C. With those examples in mind, we can ask a natural question: Are those the only examples of commuting maps? The answer will be positive if some conditions are imposed, namely, assuming that R is either a unital simple ring or a prime ring (see [3] for details). Definition 2. An additive map d : R ! R is called a derivation if d(xy) = d(x)y + xd(y) for all x; y 2 R. In 1957 E. C. Posner [14] proved the first important result regarding commuting maps. He showed that the existence of a nonzero commuting derivation d : A ! A on a prime ring A implies that A is commutative. In 1955, I. M. Singer and J. Wermer [16] proved that every continuous derivation on a commutative Banach algebra A has its range in rad(A), where rad(A) denotes the Jacobson radical of A. Thirty years later, Thomas [17] proved 3 4 that the assumption of continuity is superfluous in the result proved by I. M. Singer and J. Wermer. A natural conjecture that now appears is that Sinclair0s theorem [15] also holds without assuming continuity, that is, that every (possibly discontinuous) derivation on a Banach algebra A leaves primitive ideals (annihilator of a simple left module) of A invariant. This is usually called the noncommutative Singer-Wermer conjecture in the literature and it has a nice formulation in terms of commuting derivations (modulo Jacobson radical). In [2] M. Breˇsarhas obtained the very first results about commuting maps without any sort of assumption on the way that the map acts on the product of the elements (as in the case of derivations). One of the results that he proved was the following: Theorem 3. Let A be a simple unital ring. Let f : A ! A be a commuting map. Then f(a) = λa + µ(a), where λ is a central element of A and µ has its range in the center of A. Proof. See the original paper [1], or the survey paper [3, Corollary 3.3], or the book [4, Corollary 5.28]). It is also important to highlight that commuting maps give rise to the most basic and important example of functional identity. By replacing r by x1+x2, in the relation [f(r); r] = 0, we arrive at f(x1)x2 + f(x2)x1 − x2f(x1) − x1f(x2) = 0 for all x1; x2 2 R, which is a very simple example of functional identity. For more information or examples see for instance [4]. 1.2 Commuting Traces of Multiadditive Maps Definition 4. Let R be a ring. A map G : Rm ! R is m-additive if G is additive in each component: G(x1; : : : ; xi + yi; : : : ; xm) = G(x1; : : : ; xi; : : : ; xm) + G(x1; : : : ; yi; : : : ; xm) for all xi; yi 2 R, and i 2 f1; : : : ; mg. Definition 5. Let R be a ring and G : Rm ! R an m-additive map.
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