The work of Graham Allan was, like the man, apparently unassuming, but very influential in the international world of pure mathematics. He spent much of his career at Cambridge University, eventually becoming a Reader in Functional Analysis and Vice-Master of Churchill College.

His influence arose from his mathematical papers, frequently the seeds of new directions of research, from his beautifully presented and lucid lectures to undergraduates, and, perhaps most importantly, from the inspiration that he gave over his career to his research students.

Allan's area of research was "Banach algebras". This is a subject that is based in algebra and the standard (abstract) theory of such objects as polynomials, but which is embedded in mathematical analysis. Banach algebra theory involves a deep study of the real numbers (with some contacts to logic), and brings in both beautiful classical complex analysis and more modern abstract analysis of infinite-dimensional spaces. In this theory we posit an apparently modest connection between certain algebraic and analytic structures that form a Banach algebra; we have discovered over the years that there is actually a deep and subtle relationship between these two structures, and thus new light is thrown even on the nature of the real numbers themselves. Allan's seminal papers of the 1970s were a major step on this journey; from then on he was an international leader in the field.

Graham Allan was born in Southgate, north London, in 1936, an only child of parents who had both left school at 14. The family moved to the Cotswolds during the Second World War because his father worked in the Air Ministry, but they returned to London in 1943. Graham then attended Minchenden Grammar School in Southgate, from where he obtained an exhibition in mathematics to Sidney Sussex College, Cambridge, in 1954.

In that era, there was still compulsory military service in the UK. Although he could have postponed this service (and perhaps avoided it completely), Graham Allan served in the RAF from 1955 to 1957. This period was mainly spent at a radar station in East Anglia, and gave him a reasonable chance to study mathematics, and so he was well prepared mathematically when he started at Cambridge in 1957.

Allan was a Wrangler in Part II of the Mathematical Tripos in 1960, and completed his PhD at Cambridge in 1964 under the supervision of Frank Smithies. He became a Research Fellow at Sidney Sussex in 1963, and then began a long connection with Churchill College, being elected as a Fellow and Director of Studies in Mathematics there in 1964.

He was briefly a lecturer at Newcastle University, and then again at Cambridge until, in 1970, he was appointed Professor of Pure Mathematics at Leeds University. At that time, there was a considerable expansion of university education in the UK, and in particular the School of Mathematics at Leeds was to expand substantially, with many new faculty appointments and far more undergraduate students. The plan for pure mathematics was striking: the department was to concentrate on just three distinct areas of research in the subject. Allan was appointed to lead and build up a group in modern mathematical analysis; he was very successful in leading the development of a new undergraduate syllabus, and in building up a strong research team. He was Head of Pure Mathematics at Leeds from 1975 to 1978.

However, Allan did not welcome the increasing burden of administrative duties, coupled with the damaging financial stringency then imposed on the university, and he missed the stimulation of the very strong undergraduates and graduate students that he had had at Cambridge. He returned to Cambridge as a Lecturer in Mathematics and Fellow of Churchill in 1978; he became a Reader in 1980, again Director of Studies at Churchill from 1985, and Vice-Master of the College from 1990 until 1993.

Allan retired in 2003, but continued to teach an advanced course until 2006. A book based on these lectures will shortly be published by OUP as Introduction to Banach Spaces and Algebras. Allan's lectures were always meticulously prepared; he took great care to ensure that they were accessible to his audience, covering all details without becoming pedantic. He was always kind, quiet, thoughtful and considerate to colleagues and students, and he was an efficient organiser; he inspired deep affection in his research students and was very modest about his own considerable achievements.

H.G. Dales

Published: 18 October 2007 © The Independent