Discontinuous Homomorphisms from Banach Algebras of Operators Richard James Skillicorn, MSci A thesis submitted for the degree of Doctor of Philosophy Department of Mathematics and Statistics Lancaster University February 2016 Discontinuous Homomorphisms from Banach Algebras of Operators Richard James Skillicorn, MSci A thesis submitted for the degree of Doctor of Philosophy, February 2016 Abstract The relationship between a Banach space X and its Banach algebra of bounded operators B(X) is rich and complex; this is especially so for non-classical Banach spaces. In this thesis we consider questions of the following form: does there exist a Banach space X such that B(X) has a particular (Banach algebra) property? If not, is there a quotient of B(X) with the property? The first of these is the uniqueness-of-norm problem for Calkin algebras: does there exist a Banach space whose Calkin algebra lacks a unique complete norm? We show that there does indeed exist such a space, answering a classical open question [101]. Secondly, we turn our attention to splittings of extensions of Banach algebras. Work of Bade, Dales and Lykova [12] inspired the problem of whether there exist a Banach space X and an extension of B(X) which splits algebraically but not strongly; this asks for a special type of discontinuous homomorphism from B(X). Using the categorical notion of a pullback we obtain, jointly with N. J. Laustsen [71], new general results about extensions and prove that such a space exists. The same space is used to answer our third question, which goes back to Helem- skii, in the positive: is there a Banach space X such that B(X) has homological bidimension at least two? The proof uses techniques developed (with N. J. Laust- sen [71]) during the solution to the second question. We use two main Banach spaces to answer our questions. One is due to Read [90], the other to Argyros and Motakis [8]; the former plays a much more prominent role. Together with Laustsen [72], we prove a major original result about Read’s space which allows for the new applications. The conclusion of the thesis examines a class of operators on Banach spaces which have previously received little attention; these are a weak analogue of inessential operators. i Dedicated to the memory of Professor Charles J. Read (1958-2015) For as in Adam all die, so in Christ all will be made alive. 1 Corinthians 15:22 ii Acknowledgements The majority of my three and a half years as a PhD student have been very enjoyable, and this is due to the contribution of many different people, some of whom I will undoubtedly (to my shame) forget to mention. Firstly, I would like to thank my supervisor Dr Niels Laustsen. At the beginning of my studies I felt unsure of whether I was capable of such a substantial research project, and sometimes I have questioned my ability to keep progressing well. Your calm reassurance, gentle guidance and consistently brilliant feedback have been invaluable. Of course, your mathematical contribution to this thesis has also been significant. You took some of my partially-formed mathematical ideas and shaped them into fully-developed theorems. Thank you for giving me space to do things in my own way and for bearing patiently with my mistakes. Niels, thank you. Several others have provided helpful academic advice (often via Niels). Prof. Garth Dales and Dr Zinaida Lykova both suggested questions which have played a major part in the thesis, Dr Graham Jameson provided a clever solution to a problem in Chapter 5, and Dr Tomek Kania alerted me to a property of operator ideals on L1 in Chapter 3 and the Schreier space example in Chapter 6. I offer my thanks to each of you. I would also like to thank my examiners, Dr Martin Mathieu and Prof. Garth Dales, for their careful reading of the thesis, for an engaging and challenging viva, and for suggesting further research questions. There are many within the department at Lancaster who have been a source of friendship and encouragement. In my first year I looked up to Michał Gnacik and Tomek Kania in particular. Thank you both for taking time to get to know me and sharing your wisdom. It has been great to have Mateusz Jurczyński alongside me all the way: your vast knowledge and good humour have made PhD life interesting. To my officemates, especially Caroline Brett and Henry Chan, and latterly Cindy Xue, Nifeen Altaweel, Konrad Krolicki and Jess Welding, thanks for sharing the postgrad experience with me. Some of our chats I will never forget. There are many others to thank including, but not limited to: Ghada Badri, Thomas Booker-Price, Gareth Case, Sean Dewar, Alec Gullon, Jason Hancox, Bence Horvath, Heather Johnson, Lefteris Kastis, Wentao Li, James Maunder, Chris Menez, Ben Pickering, Hattie Serocold, David Sudell, Shane Turnbull, Chigozie Utazi, Ian Wadsworth, Hugo Winter, Jared White and Ping Zhong. Amongst the lecturing staff there have been many who have been generous with their time and shown interest in my work; I must mention Dr Alex Belton, Dr Yemon Choi, Prof. Garth Dales, Dr Robin Hillier, Dr Derek Kitson and Prof. iii Martin Lindsay. I would also particularly like to thank Prof. Stephen Power for reading my thesis and for providing a stimulating and helpful ‘mock viva’. I wish the best for all of you. A great pleasure of the PhD has been the opportunity to travel and meet new people at conferences. I have met too many good people to mention, but let me extend my thanks to those who made me feel welcome. On a practical note, I acknowledge with thanks the funding provided by the Lancaster University Faculty of Science and Technology during the course of my studies, including a bursary for travel. Academic life can be a bubble; I am grateful to those who have frequently reminded me of the outside world. To my parents and brother, thank you for always encouraging me to do what I love, and for your constant support. To my many friends at Moorlands Church, I am grateful that you continually point me to what really matters in the world. Last, but certainly not least, I must thank my wife Erin. I could not have completed my studies without your constant support, love and encouragement. Thank you for your wisdom and your ability to make a problem seem infinitely smaller with a few thoughtful words. Thank you for your interest in my work despite its technical nature. I think you are now better at explaining it to the layman than me! Much of the work in this thesis has been influenced by that of the late Professor Charles Read, formerly of the University of Leeds. I have spent many hours poring over his work, appreciating its brilliance. Of course I have only actually examined a very small part of his significant body of research, but nevertheless, I feel I have got to know him mathematically. I also had the pleasure of meeting him on a couple of occasions. Although I cannot say I got to know him well, he demonstrated a real interest in what I had done and spoke kindly to me. I was greatly saddened to hear of his sudden death in August 2015. Only the week before I had been speaking to him at Banach Algebras and Applications 2015 in Toronto, Canada, and I was shocked to hear of his passing so soon after. Given its content, it seems right to dedicate the thesis to his memory. iv Declaration The research presented in this thesis has not been submitted for a higher degree elsewhere and is, to the best of my knowledge and belief, original and my own work, except as acknowledged herein. Chapter 2 contains published work [101], which is reproduced, in a modified form, with permission. Chapter 3 and Chapter 4 contain published joint work with N. J. Laustsen [71]. Chapter 5 (specifically Section 5.4) contains joint work with N. J. Laustsen [72] (submitted). v Contents 1 Introduction 1 1.1 Thesis overview . .1 1.2 Preliminaries . .3 1.3 Introduction to Read’s Banach space . .7 2 Uniqueness-of-norm for Calkin Algebras 12 2.1 History of the problem . 13 2.2 Calkin algebras with non-equivalent complete norms . 17 3 Splittings of Extensions of Banach Algebras 23 3.1 The general theory . 24 3.1.1 Splittings of extensions of B(X) ................ 29 3.2 Extensions and pullbacks of Banach algebras . 32 3.2.1 Pullbacks in the category of Banach algebras . 32 3.2.2 The connection to extensions . 35 3.3 Extensions which split algebraically but not strongly . 39 3.3.1 The pullback method . 39 3.3.2 An alternative method . 41 3.3.3 Extensions of algebras of operators . 47 4 Homological Bidimension 50 4.1 Cohomology groups . 51 4.2 Calculating second cohomology groups and homological bidimension 54 4.3 Weak bidimension . 64 5 Read’s Banach Space ER 68 5.1 Preliminary results . 69 5.2 The construction of ER ......................... 72 5.2.1 James-like spaces . 72 5.2.2 Lorentz sequence spaces . 77 5.2.3 Direct sums of Banach spaces and the space ER ....... 86 5.3 Analysing the space ER ........................ 90 vi 5.3.1 Summary of Section 3 of Read’s paper . 98 5.3.2 The central lemma . 99 5.4 The algebra of bounded operators on Read’s space . 111 5.5 Commutators on ER .......................... 117 6 Weakly Inessential Operators 121 6.1 The operator ideal of weakly inessential operators .