# Gerard John Murphy

### Quick Info

Drimnagh, Dublin, Ireland

Cork, Ireland

**Gerard Murphy**was an Irish mathematician who worked in functional analysis.

### Biography

**Gerard Murphy**'s parents were Laurence Murphy (known as Larry) and his wife Mary (known as May). Larry Murphy, who died on 22 October 1987 at the age of 82, was a window cleaner. Gerard had seven younger siblings, two brothers and five sisters: John, Derek, Rita, Linda, Joan, Carol and Lauren. Gerard attended the school Our Lady of Good Counsel in Mourne Road, Dublin. He left school in 1963 when he was fourteen years old so that, as the eldest child, he could earn money to help his parents support the large family.

The fourteen year old Murphy's first job was as a telegram boy making deliveries from the Post Office in O'Connell Street, Dublin. However, although this brought in some money, he did not find it satisfying and he suggested to his father that he might prefer cleaning windows. His father was happy to see him join his window cleaning business but this still did not give him the satisfaction that he was looking for so, continuing to clean windows, he began reading books whenever the opportunity arose. He was fortunate since the Simms family were one of his father's customers and they were happy to lend Murphy books from their collection. David John Simms had studied mathematics at Trinity College Dublin, graduated in 1955 and was on the staff of Trinity College Dublin from 1964. David Simms was the nephew of George Otto Simms, who was Archbishop of Dublin. The Simms family lent Murphy an encyclopaedia consisting of several volumes which he read avidly. He was not the only one of the family to take up reading books since his example led to all his younger siblings also finding a love of learning from books.

After five years working as a window cleaner and studying in his free time, Murphy made the decision to give up work and concentrate full time on studying at home. As one might imagine, this did not go down well with other members of his family, but Murphy showed great determination for following the course of action on which he had decided. His aim was to study engineering at Trinity College Dublin but, having no qualifications, he decided to sign up with the International Correspondence Schools and take A-level courses in Mathematics and Computer Science. Murphy was not studying completely on his own, however, for he had occasional help from David Simms. When he felt confident in his abilities, Murphy had to make a journey to England so that he could sit the examinations in London. This was the first time that the young man had been outside Ireland but he performed well in the examinations passing the necessary A-levels to gain entrance to Trinity College, Dublin.

Murphy was still aiming to study engineering at university but this was a course which was oversubscribed at Trinity College and the university made decisions on which students to accept based mainly on secondary school reports. Since Murphy had never attended a secondary school he had no school report and his application for engineering was rejected. This was not a total disappointment to Murphy who, by this time, had developed a love for mathematics. He therefore asked to be considered for entry to the honours mathematics course at Trinity College. However, he did not have passes in English or a foreign language so, despite a very strong recommendation from David Simms, the admissions officer decided that all Trinity could offer to Murphy was entry to a general degree. Murphy accepted, despite having to take courses in pure and applied mathematics well below the level that he was at, and entered Trinity College Dublin in October 1970.

Although Murphy was signed up for a general degree, the Professor of Mathematics, Brian Hughes Murdoch, allowed him, in addition to the courses for the general degree, to attend the course for honours students. Murdoch, who had studied for his Ph.D. at Princeton with William Feller, also allowed Murphy to take the examination at the end of the first term. Murphy performed so well in the examination that he was allowed to transfer from a general degree to an honours degree in January 1971. His performance was outstanding and he was awarded a Foundation Scholarship which covered his fees and his board and lodging. He graduated with a B.A. in 1974 with a First Class honours degree in mathematics and was awarded the Gold Medal. Churchill College, University of Cambridge, awarded Murphy a Gulbenkian Research Studentship so he was able to matriculate at Churchill College in October 1974 to begin studying for a Ph.D.

At Cambridge, Murphy was advised by George A Reid and he worked on non-Archimedean functional analysis. Finbarr Holland explains the background in [1]:-

The theory of non-Archimedean Functional Analysis was begun in the 1940s, and, in the succeeding decades, efforts were made to extend the standard theorems of classical Functional Analysis by replacing the underlying field of real or complex numbers with a non-Archimedean field ... The standard example of such a field is provided by the p-adic numbers, and, no doubt, this served to motivate the study of other algebraic structures over a non-Archimedean field.The study of C*-algebras was an important topic and a natural step was to try to put these into a non-Archimedean setting. Before Murphy began his research a number of attempts to do this had been made but they had not been too successful. George Reid suggested to Murphy when he began his research that he try to develop a more successful theory of non-Archimedean C*-algebras. Murphy was soon making excellent progress and, in his second year of study at Churchill College, he was awarded the Knight Research Prize for an essay he had written. Murphy was awarded a Ph.D. in 1977 for his thesis

*Non-Archimedean Banach Algebras*. He published his first paper,

*Commutative non-Archimedean C*-algebras*, which was submitted in 1976 and appeared in 1978. This major 33-page paper contains results from his thesis [1]:-

... already in this paper one can discern early signs of his ability to present difficult ideas in a clear and cogent manner, a skill which was another of his hall-marks. Aside from this, moreover, one learns from his thesis his penchant for algebraic methods and axiomatics, his sense of mathematical aesthetics, his ability to deal with abstract concepts, and his knowledge and understanding of several different areas of Algebra, Topology and Functional Analysis, skills which he displayed in abundance later in the seventy or so research papers he subsequently wrote.After completing his research at Churchill College, Cambridge, Murphy returned to Dublin when he was awarded a Government Postdoctoral Research Fellowship. He continued to undertake research at Trinity College, but he also did some teaching there. While still at Cambridge, Murphy had discussed various ideas with Trevor West (Timothy Trevor West, 1938-2012) and Roger Smyth, who were on the staff of Trinity College, and Bruce Barnes, who was on the faculty at the University of Oregon. Barnes spent a sabbatical year at Trinity College Dublin and, when Murphy returned to Dublin he became a member of this team. Their research led to the monograph

*Riesz and Fredholm theory in Banach algebras*which was published in 1982. The four authors state that their aim is:-

... to highlight the interplay between algebra and spectral theory which emerges in any penetrating analysis of compact, Riesz and Fredholm operators on Banach spaces.In a review of the book, Harro Heuser writes:-

Their little book proves, among other things, that they have fully (and beautifully) achieved this aim.Before this book was published, however, Murphy had published four joint papers with Trevor West, namely

*Spectral radius formulae*(1979),

*Decomposition algebras of Riesz operators*(1980),

*Removing the interior of the spectrum*(1980), and

*Decomposition of index-zero Fredholm operators*(1981). For a list of Murphy's papers see THIS LINK.

In 1980 Murphy, supported by Peter Fillmore from Dalhousie University, Halifax, Canada, was awarded a two-year fellowship from the Canadian Government. This enabled him to spend two years at Dalhousie where he collaborated with Peter Fillmore and Heydar Radjavi, both of whom had made major contributions to the study of C*-algebras. In 30 November 1981, while in Dalhousie, Murphy submitted his paper

*Hyperinvariant subspaces and the topology on Lat, A*for publication. It appeared in print in 1984. Here is Murphy's summary:-

The lattice of invariant subspaces of an operator is a metric space. We give various topological conditions on a point in the lattice which ensure it is a hyperinvariant subspace for the operator.Also while working in Dalhousie he wrote the joint paper

*Associative and Lie subalgebras of finite codimension*(1983) with Heydar Radjavi. During his second year in Dalhousie, Murphy married Mary O'Hanlon; they had four children: Alison Murphy, Adele Murphy, Neil Murphy and Elaine Murphy. After spending two years in Dalhousie, Murphy moved to the United States where he worked for a year as an Associate Professor at the University of New Hampshire, Durham, followed by a year at Oregon State University. At Oregon he was able to work again with Bruce Barnes. While at the University of New Hampshire he wrote the paper

*Lie ideals in associative algebras*(1984). In this paper he showed how, in a certain large class of algebras, one can associate with each Lie ideal a corresponding associative ideal. This association helps in the study of Lie ideals, and is especially useful for studying simple algebras. While in North America he made contact with C K Fong and they wrote two joint papers:

*Averages of projections*(1985) and

*Ideals and Lie ideals of operators*(1987). In the second of these papers both Fong and Murphy give their address as the University of Toronto.

In 1984 Murphy returned to Ireland where he was appointed as a lecturer in Mathematics at University College, Cork. He would work at Cork for the rest of his life. He wrote a number of important survey articles such as

*Extensions and K-theory of C*-algebras*(1987) and

*Toeplitz operators*(1989). He summarises the paper on Toeplitz operators, a favourite topic for Murphy, as follows:-

There are few classes of operators on a Hilbert space about which one has very detailed information, apart from the normal operators and the compact operators. An exceptional class about which much is known is the class of Toeplitz operators. This paper gives a brief survey of some aspects of their theory, from its origin near the beginning of this century to the present day.His most important survey, however, was his book

*C*-algebras and operator theory*(1990). Murphy writes in the Preface:-

This book is aimed at the beginning graduate student and the specialist in another area who wishes to know the basics of this subject. The reader is assumed to have a good background in real and complex analysis, point set topology, measure theory, and elementary functional analysis.E Gerlach writes in a review:-

The author has succeeded in crafting a concise, yet highly accessible introduction to the theory of C*-algebras. The core material of the subject is well covered, and a number of topics are taken up which have seen much research activity in recent years (in particular the theory of C*-tensor products, and K-theory). Some topics from operator theory, which are needed for the general exposition or are involved in some of the examples and applications, are also included in the exposition. ... The book is not intended to be encyclopaedic; the author has chosen his material well, and the exposition is excellent. Working through this book and learning C*-algebras from it should be fun and very rewarding for any student.Holland describes Murphy's teaching in [2]:-

He took his teaching responsibility seriously and sought to inculcate the notions of precision and proof handed down by the ancient Greeks. He designed his courses to further the student's understanding and appreciation of mathematics, not only as a tool for understanding other disciplines, but also as one of mankind's greatest scientific and cultural achievements.More details about Murphy as a teacher are given in [1]:-

He put a lot of thought and preparation into his courses, which were designed to reflect his own approach and ideas in terms of selection of material, examples, homework and student motivation. Never content to use a colleague's lecture notes, he always designed his own. These were models of clarity and precision, and are as fresh and novel today as they were when he delivered them.In 1992 he was promoted to statutory lecturer at Cork, he was elected to the Royal Irish Academy, and in 1995 he was promoted to associate professor:-

... in recognition of the quality and quantity of his research output, the calibre of his teaching, and the overall contribution he made to the running of the Department of Mathematics and the well-being of the College.His success in obtaining funds from the EU enabled him to support conferences, to fund postgraduate students and have postdoctoral assistants. The first of his international conferences on 'Operators Algebras' was held at University College, Cork in 1995. He continued to publish interesting papers such as

*The analytic rank of a C*-algebr*a (1992) which he describes as follows:-

A number of concepts of rank have been proposed recently. An important part of the motivation is to have an analogue of topological dimension for a C*-algebra. Topological stable rank was introduced by M A Rieffel (1983); it was later shown to be identical to Bass stable rank. Following similar lines, Brown and Pedersen proposed the real rank as a useful analogue of topological dimension. In this paper, the author presents another concept of rank for a C*-algebra, the analytic rank which is based on a theory for abelian normed algebras and which reduces to the dimension of the spectrum in the abelian case.In 1992 he presented a paper to the 'Functional analysis and operator theory' Conference in Warsaw which surveyed:-

... some aspects of the theory of derivations on Banach algebras. No attempt is made at completeness; rather, our intention is to cover the basic theory and to discuss some recent results.Another conference held in Warsaw in 2001 was on the topic of 'Noncommutative geometry and quantum groups'. Murphy discussed recent work on twisted graded traces, an extension of Connes's cyclic cohomology, invariant linear functionals on covariant calculi and the Hodge, Dirac and Laplace operators in this setting.

He became Head of the Mathematics Department at Cork in 1999 and continued in this role for five years. In the summer of 2005 he organised his final international conference on C*-algebras in Cork. In the autumn of that year he was diagnosed with cancer of the colon and liver. In May 2008 the conference 'Operator Theory and Operator Algebras in Cork' was held in his memory. The announcement stated:-

Gerard John Murphy was a lecturer and professor of Mathematics at University College Cork from 1984 until his untimely passing in October 2006. ... A three-day conference will be held in the National University of Ireland, Cork from 7-9 May, 2008 focusing on operator theory and operator algebras, the two areas in which Gerard made major contributions. There will be plenary talks by two principal speakers and a number of invited talks by other participants, the emphasis being on modern developments in these fields.As to Murphy's interests outside mathematics we quote from [1]:-

Gerard was very widely read, and delved deeply into History and Economics, especially. Indeed, he had every intention, apparently, of writing an Economic History of Ireland, and had written copious notes in print form - which was his style of writing - which he hoped to pull together in book form at some stage. Another plan of his was to write children's stories, many of which he composed for his own children, of whom he was exceedingly proud.As a final comment we note that Murphy's 35-page paper

*Representation and index theory for Toeplitz operators*was published in the

*Transactions*of the American Mathematical Society in 2010, four years after his death. The paper gives no indication that this was a posthumous publication and one can only assume that he submitted it himself since the paper still gives his Cork address.

### References (show)

- F Holland, Gerard J Murphy (1948-2006),
*Irish Math. Soc. Bulletin***59**(2007), 9-27. - F Holland, Mathematician who rose to the top of his profession,
*The Irish Times*(Saturday 28 October 2006).

### Additional Resources (show)

Other pages about Gerard Murphy:

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Written by J J O'Connor and E F Robertson

Last Update February 2016

Last Update February 2016