Died: 9 August 2007 in Cambridge, England

**Graham Allan** was born in Southgate, north London. He was [1]:-

... an only child of parents who had both left school at14.

When World War II began in 1939, Graham was only three years old. His father worked for the Air Ministry and soon the family were moved to the Cotswolds where they spent four years before returning to London in 1943. Graham attended Minchenden Grammar School in Southgate and his outstanding mathematical achievements led to him being awarded an exhibition in mathematics to Sidney Sussex College, Cambridge, at the age of eighteen. He chose to undertake military service at this stage rather than postpone it until after his university studies which was the route taken by many university students.

For the two years 1955-57 Allan served with the Royal Air Force, spending most of the time stationed in East Anglia at a radar station. These were not years in which he neglected his mathematical interests, however, rather they gave him time in which he was able to prepare himself very well for his Cambridge studies. He entered Sidney Sussex College in 1957, took Part II of the Mathematical Tripos in 1960 and Part III in the following year. He then continued to undertake research supervised by Frank Smithies and, after being appointed as a Research Fellow, Sidney Sussex College in 1963, he was awarded a Ph.D. in the following year for his thesis *Contributions to the theory of locally convex spaces*. During his time as a research student he married Elizabeth Gemmell in 1962; they had two daughters Juliet and Clare.

After completing his doctorate, Allan was elected a Fellow and Director of Studies of Churchill College, Cambridge. He was three years in this position, from 1964 to 1967 and during this time he lectured to Garth Dales who writes [2]:-

I took Graham's Part III course on Banach algebras in the year1966 - 67. I was very much attracted by the beautiful clear lecturing style which covered the details carefully, but never became pedantic; he always took great care to ensure that his lectures were accessible to his audience. The material was, to me, a lovely blend of algebraic foundations with a substantial super-structure of real, complex, and functional analysis.

In 1967 he was appointed a Lecturer in Pure Mathematics at Newcastle University. He spent only two years there before returning to Cambridge as a Lecturer in Pure Mathematics. After a year back at Cambridge he was offered a professorship in Pure Mathematics at Leeds University in 1970 [1]:-

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.

From 1975 to 1978 he served as Head of Pure Mathematics at Leeds. Garth Dales writes [1]:-

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 in1978...

In order to gain further understanding of why Allan wished to return to Cambridge, we should look at this point at his leading mathematical contributions. He began publishing papers shortly after completing his doctoral studies, his first papers being naturally based on the material in his thesis. He published *A note on B*-algebras* (1965), *A spectral theory for locally convex algebras* (1965), *On a class of locally convex algebras* (1967), *On one-sided inverses in Banach algebras of holomorphic vector-valued functions* (1967), and *Holomorphic vector-valued functions on a domain of holomorphy* (1967). He continued to publish two papers a year until 1971, these papers making deep and significant contributions to Banach algebras. He was invited to contribute a survey article to the *Bulletin* of the London Mathematical Society and this survey *Some aspects of the theory of commutative Banach algebras and holomorphic functions of several complex variables* was published in 1971:-

In this survey article the author outlines those parts of holomorphic function theory(in particular, the holomorphic functional calculus)that have been applied in the study of commutative Banach algebras, specifically excluding topics that are peculiar to uniform algebras, where the applications have been most extensive. Most of the material has become well known during the last few years but the author does touch on some very recent results and open problems. He concludes with a good bibliography of71titles.

However, administrative duties at Leeds had a significant impact on the number of his publications. In the five years 1972 to 1976 inclusive he published four papers, one being *Several complex variables and Banach algebras* in 1976:-

This paper is a nicely-written brief introduction to that part of the general theory of commutative Banach algebras in which complex function theory plays a significant role. The author assumes no previous knowledge of Banach algebra theory, but some acquaintance with the beginnings of functional analysis is assumed as is the general notion of a holomorphic function of several complex variables. The main object of the exposition is the construction of the holomorphic functional calculus in several variables and the application of this calculus to the Silov idempotent theorem, the local maximum modulus theorem and the Arens-Royden theorem.

Between 1976 and 1983 only one of Allan's papers was published, so it become easy to understand how he wished to return to an environment where he could concentrate more on the research he loved. Having moved from a professorship to a lectureship in 1978, he became a Reader in 1980. In 1985 he was again appointed Director of Studies at Churchill, and then Vice-Master of the College from 1990 until 1993. He retired in 2003. Garth Dales writes:-

The work of Graham Allan was, like the man, apparently unassuming, but very influential in the international world of pure mathematics. ... 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.

In fact Allan had over 20 research students, including include John Rennison (Kent), Ian Craw (Aberdeen), Garth Dales (Leeds), Peter McClure (Manitoba), Peter Dixon (Sheffield), Ghotsi Haghany (Isfahan), Thomas Ransford (Laval), Michael White (Newcastle), Frederic Gourdeau (Laval) and Thomas Vils Pedersen (Copenhagen).

In [1] it is noted that:-

Graham became a Roman Catholic in1979, persuaded by intellectual considerations; he deeply loved philosophy and theology, and this was a major part of his life. ... Always Graham was kind, quiet, thoughtful, and considerate; he inspired great affection in his research students and others; he was very modest about his own achievements. His former students and many friends in the mathematical community will miss him greatly.

**Article by:** *J J O'Connor* and *E F Robertson*

**July 2008**

[http://www-history.mcs.st-andrews.ac.uk/Biographies/Allan_Graham.html]