Nigel Kalton’s Work in Isometrical Banach Space Theory

Nigel Kalton was one of the greatest mathematicians of the last 40 years, although he did his best to conceal this fact. An outsider wouldn’t recognise the mathematical giant that he was in this modest person who was always friendly and good-humoured and who was more than willing to share his ideas with everyone. Nigel published more than 260 papers (including several books) not only in Banach and quasi-Banach space theory, but also in so diverse fields such as game theory, continued fractions, harmonic analysis, operator semigroups and convex geometry. Every single of these papers contains a deep contribution by Nigel, often taking care of the most difficult case that his coauthors would have to leave open without his help. He had a wide interest in mathematics, and his problem solving abilities were legendary. For instance, once after a colloquium talk on continued fractions he got hooked on the subject and redeveloped the theory for himself over one weekend, eventually solving the problem exposed in the talk. I met Nigel for the first time at the conference on Banach spaces in Mons in 1987. It so happened that on the day after the conference we were both waiting for the same train to Paris, but not for the same coach: he told me that he always rides the first class, adding, “I’m snobbish.” Of course he couldn’t be more wrong in his self-assessment! Some years later he solved a big problem in M -ideal theory (see Section 2), a problem, where we, a group of fresh Ph.D.s in Berlin, couldn’t get anywhere. He emailed me a file with his solution, and this was the beginning of our collaboration, in which more often than not I felt like a pedestrian next to a racing-car. In June 2010, after a talk at the conference in Valencia with a somewhat set-theoretical flavour, I reminded him of the quote from Star Trek, “It’s mathematics, but not as we know it.” I knew that this would strike his sense of humour; I did not know that this would be the last time I saw him. In the next few sections I will try to survey some of Nigel’s contributions to Banach space theory. I will restrict myself to problems of an isometric nature, but even this narrower area is still so rich that omissions and misconceptions will be inevitable. Certainly, the only way to do justice to Nigel’s genius would be to not only paraphrase the main results, but to expound all the ideas contained in his papers. I have to leave this to an abler mathematician.