by George W. Mackey
Marshall Stone made major contributions to functional analysis that both shaped the field and have guided it for many decades. His first major contribution was to the spectral theory of unbounded self-adjoint operators on Hilbert space. The spectral theory of bounded self-adjoint operators had been established nearly two decades earlier by the work of Hilbert, Hellinger and Hahn, but that work excluded the case when the operator was unbounded. Unbounded operators provide some of the most important examples and potential applications of spectral theory, such as ordinary or partial differential operators, and in particular the differential operators that appeared in the then new quantum mechanics. Stone had worked on eigenfunction expansions (i.e., a form of spectral theory) for ordinary differential operators in his dissertation and in his early work up to 1928, so he was primed to attack this spectral theory problem in general. He did so starting in 1929 and in grand style, first announcing his results in three notes in the Proceedings of the National Academy of Science [4], [5] and then publishing details in a magisterial 600 page book [7] that has remained a classic in the subject. The third of these PNAS notes is worthy of special notice as the results here are of particular importance for both classical and quantum mechanics and were left out of the 1932 book solely for lack of space. This paper contains a theorem on one parameter groups of unitary operators, known subsequently as Stone’s theorem, as well as a theorem of Stone that was also independently proved by von Neumann [e4], which asserts the unicity of a finite set of self-adjoint unbounded operators satisfying the classical commutation relations of quantum mechanics and acting irreducibly. This result is universally known now as the Stone–von Neumann theorem.
After his work on unbounded operators Stone turned his attention to Boolean algebras and announced his new results in two PNAS notes and [9], with the details appearing in three lengthy papers [10], [11], [12]. These papers revolutionized the study of Boolean algebras and made the subject part of functional analysis; for instance Stone showed that any Boolean algebra could be realized as an algebra of subsets of its maximal ideal space, now called the Stone space. These papers also contained important applications to topological spaces including what is now called the Stone–Weierstrass theorm, which is a vast generalization of the 19th century theorem of Weierstrass on polynomial approximation. The Stone–Weierstrass theorem is one of the basic tools of functional analysis. His results also include analysis of the different ways in which a topological space may be be embedded in a compact space (or be compactified) and to what is now called the Stone–Cech compactification. Two more notes in the PNAS [13], [14] address the subject of spectra in general, and which include both his theory of self-adjoint operators and his theory of Boolean algebras. After the war, Stone turned his attention to a form of axiomatic integration and published four notes in the PNAS [15], [17], [16], [19]. In addition Stone published a number of expository articles, interpreting and explaining many of his previous results. He also turned his interests to thinking and writing about the nature and philosophy of mathematics and to the teaching of mathematics. [21] and [22] are prime examples of his thoughtful and insightful thinking and writing on these subjects.
Marshall Harvey Stone was born in New York City on April 8, 1903 to Harlan Fiske Stone and Agnes Harvey Stone. Marshall had a younger brother, Lauson Harvey Stone born in 1904, who became a lawyer. Harlan Stone was a prominent lawyer, legal educator, and jurist. Born in Chesterfield, New Hampshire, Harlan grew up on his family’s farm near Amherst, Massachusetts, with all the usual boyhood farm chores. The family had moved to Amherst in order to improve the educational opportunities for their children. Harlan came from a long line of New England forebears, the first being Simon Stone, a Puritan who emigrated from England and settled in the Massachusetts Bay Colony in 1635. As Mason [e10] put it, Harlan “never doubted that the rugged qualities of the Yankee hill farmer were born and bred in him.”
Harlan attended Amherst public schools and then entered Amherst College where he excelled, graduating Phi Beta Kappa in 1894, serving as class president for three years, and as a stalwart on the football team. After a year of teaching in high school, he decided on a career in law and won admission to the Columbia School of Law, supporting himself by part-time teaching in history at Adelphi Academy in Brooklyn. He did well in law school and entered law practice in New York City in 1898. Shortly afterward he was offered a part-time teaching position at Columbia School of Law, and in 1910 he was offered and accepted the position of Dean of the Columbia School of Law. He held this position until 1923 when he resigned and returned to the full time practice of law as head of litigation for the leading Wall Street firm of Sullivan and Cromwell. But not for long, as his friend from college days at Amherst, President Calvin Coolidge, appointed him in 1924 as Attorney General of the United States. Stone was seen as a man of unimpeachable integrity, and was given the job of cleaning up the scandals in the government from the previous administration. His nomination was greeted with both relief and approval, and the New York Times editorialized that “In character, in legal attainments, in approval of his brethren of the law, and physical vigor and ability to cope with the severe labor which will be laid upon in the Department of Justice, Mr. Stone is unusually well qualified.” Then in 1925, Coolidge appointed him as Associate Justice of the Supreme Court, where during the 30s, Stone together with Justices Brandeis and Holmes, (and later with Holmes’ replacement, Justice Cardoza) made up the liberal wing of the Court. In 1941 President Roosevelt appointed Stone, a life-long republican, as Chief Justice of the Court, a position he held until his death in 1946 [e10], [e14].
Agnes Harvey Stone was also born in Chesterfield, New Hampshire, coming from a similar background as Harlan. The Stone and Harvey families were good friends and often visited each other even after the Stones moved to Amherst. Harlan and Agnes were childhood companions and became childhood sweethearts, marrying in 1899. In addition to her responsibilities as wife and mother, Agnes was an accomplished painter, specializing in watercolor landscapes. She had a number of shows of her work including at the Corcoran Gallery in Washington D.C. which in 1937, staged a show featuring 24 of her works, and staged a second show in 1941 featuring 30 of her works. Finally in 1945 the Virginia Museum of Fine Art staged a show of sixty-four of her works [e10].
While Harlan Stone was based in New York, the family lived in Englewood, New Jersey, and Marshall and his younger brother Lauson attended the local public schools. The family also built a summer home on the Isle au Haut, an island in Penobscot Bay in Maine. The family enjoyed their summer vacations there and found it very relaxing.
As a young boy, Marshall often accompanied his father to Columbia University sitting quietly and observing his father in office and classroom. He and his brother had household chores assigned by their father performed for “modest” remuneration. Marshall was a vigorous and healthy lad, playing football, baseball and tennis, and sailing small boats at their summer house in Maine [e12]. In 1919 at the age of only 16, Marshall entered Harvard University from which he was graduated, summa cum laude, in 1922.
Although it had been assumed that Marshall would follow his father into the law, a growing fascination with mathematics led to an extraordinary arrangement in which he spent the academic year 1922–1923 as a part-time instructor at Harvard to find out whether he liked teaching. It turned out that he did and he proceeded quickly to write a Ph.D. thesis under the direction of G. D. Birkhoff. The degree was awarded in 1926 but the work was completed rather earlier. The very distinguished mathematical career of Marshall Stone was under way. Before settling down at Harvard for the thirteen-year period 1933–1946 Stone held a variety of positions. He was at Columbia from 1925 to 1927, at Harvard from 1927 to 1931, at Yale from 1931 to 1933 and at Stanford for the summer of 1933. He became a full professor at Harvard in 1937. These early years were enormously fruitful ones for Stone’s career as a research mathematician — so much so that he was elected a member of the National Academy of Sciences in 1938 at the unusually early age of 35.
His first paper was a short note on normal orthogonal sets of functions published in 1924 and by 1928 had published a number of additional papers on various aspects of the theory of orthogonal expansions — special emphasis being placed on expansions in terms of eigenfunctions of linear differential operators. See for instance [1], [3], [2]. This was one of the principal interests of G. D. Birkhoff and Stone’s work was in the same tradition. Then in 1929 he began to work on the abstract theory of possibly unbounded self-adjoint operators in Hilbert space, announcing his results with three notes published in the Proceedings of the National Academy of Sciences [4], [4], [5]. This work culminated in a six hundred page book which is now one of the great classics of twentieth century mathematics. It was entitled “Linear transformations in Hilbert space and their applications to analysis”. This comprehensive and beautifully written book [7] has been enormously influential. Modern functional (or abstract) analysis began with the ideas of Volterra on “functionals” in the late nineteenth century and was transformed and given considerable impetus in the first two decades of the twentieth century by the work of Hilbert and F. Riesz. The very different book of Banach [e6] together with Stone’s book set the stage for the extensive developments since that time. Stone’s father, who was now on the Supreme Court, was very proud of his son’s achievements in mathematics, and one of his law clerks reported that the justice always kept a copy of his son’s book on his desk in the Supreme Court [e12].
In his introduction Stone freely acknowledges his scientific debt to J. von Neumann. Von Neumann published a long paper on the same subject [e3], and it is not easy to disentangle their respective contributions. What is clear is that Stone was originally stimulated by preliminary work of von Neumann but had many key ideas quite independently. Moreover the whole last half of the book, including the chapter on spectral multiplicity theory and the extensive applications to differential and integral operators, has no counterpart in von Neumann’s writings. The central point of the work of both men was the extension of Hilbert’s spectral theorem from bounded to unbounded operators. This extension was made necessary by the problem of making mathematically coherent sense of the newly discovered refinement of classical mechanics known as quantum mechanics. Here an important part of the problem was discovering the ”correct” definition of self-adjointness for unbounded operators. This correct definition is rather delicate and the extension of the older theory of Hilbert and others was a major task.
The last of the three notes mentioned above was entitled ”Linear transformations in Hilbert space III. Operational methods and group theory”. The material it summarized was originally meant to be included as an extra chapter of the book but was omitted for reasons of space. The two theorems it announces are of sufficient importance to be discussed here in some detail.
Three years earlier, in 1927, Hermann Weyl [e1] and Eugene Wigner [e2] had introduced group theoretical methods into the new quantum mechanics in quite different ways. Weyl’s idea was to use group theory to help clarify the foundations. His paper, written in physicists’ language, implicitly conjectured two theorems about one-parameter groups of unitary operators in Hilbert space. Stone’s note states these conjectures as carefully formulated theorems, announces that he is in possession of proofs and gives some indication of their nature. See [6] for his proof and also [e7]. Both theorems were not only important for quantum mechanics, in the manner indicated by Weyl, but were also highly significant early steps in the then nascent unitary representation theory of non compact locally compact groups. One of them also played an important role in the chain of events leading through a note of B. O. Koopman [e5] to the ergodic theorems of von Neumann and Birkhoff and on to modern ergodic theory.
At this point it is useful to distinguish between two versions of one of Stone’s theorems. The version suggested by Weyl’s paper (and which stimulated Koopman) asserts that for every one parameter unitary group \( t \leq Ut \) there is a unique self adjoint operator \( H \) such that \( Ut = e^{l H t} \). The version emphasized in Stone’s note is an analogue of the spectral theorem for one parameter unitary groups. This version has the great advantage that it can be generalized almost verbatim to arbitrary (separable) locally compact commutative groups.
The other theorem — the celebrated Stone–von Neumann uniqueness theorem — states the uniqueness of the irreducible solutions of the Heisenberg commutation relations in integrated form see; also [e7]. This result may be interpreted as giving a complete determination of all irreducible representation of a certain non-compact non-commutative locally compact group — now well known as the Heisenberg group. So interpreted, it is the first example of such a determination by about a decade. Finally a series of natural generalizations of the Stone–von Neumann uniqueness theorem culminated in the imprimitivity theorem and the extension of the notion of induced representation from finite groups to general locally compact groups.
Shortly thereafter Stone published two more notes in the Proceedings of the National Academy [8], [9] which seemed at first to represent a completely new departure. They were entitled “Boolean algebras and their applications to topology” and ”Subsumption of Boolean algebras under the theory of rings”. Actually, just as Stone’s work on spectral theory may be regarded as a natural outgrowth of his earlier work on concrete eigenfunction expansions, so can his work on Boolean algebras be regarded as a natural outgrowth of his work on spectral theory. This is because of the role played in spectral theory by Boolean algebras of projections. In an entirely characteristic attempt to get to the bottom of things Stone undertook a thoroughgoing study of Boolean algebras and made a number of far reaching discoveries relating Boolean algebras to general topology on the one hand and to the theory of rings and ideals on the other.
The discovery of these connections has had significant consequences for all three subjects. One beautiful result is the celebrated Stone–Weierstrass theorem vastly generalizing the theorem of Weierstrass concerning the uniform approximability of arbitrary continuous functions on a finite interval by polynomials. Another is the natural one to one correspondence between all compact Hausdorff spaces on the one hand and certain rings on the other. Stone’s studies of the relationship between compact spaces and rings of continuous functions anticipated important elements in the modern theory of commutative Banach algebras. Another noted result was what became known as the Stone–Cech compactification of a topological space that was discovered independently by Stone and Eduard Cech in 1937 [e8]. The detailed development of Stone’s ideas on Boolean algebras, and general topology. were published in three lengthy papers [10], [11], [12], respectively. Applications of these new ideas to spectral theory were announced in notes published in [13] and [14], with some details appearing in [18].
Soon after the entry of the United States into World War II the character of Stone’s work underwent a considerable change. For several years he was engaged in secret work for the U.S. government, serving as a consultant for the U.S. Navy, 1942–43 and then in the office of the Chief of Staff of the Army, 1943–45. As part of his duties, he was sent on classified missions to China, Burma, and India as well as the European theater.
The year after the end of the war he resigned his position at Harvard to take on the chairmanship of the mathematics department at the University of Chicago. This once great department had been declining in quality and Stone’s mission was to strengthen it and bring it up to its former stature. In this he succeeded admirably. Before very long it was regarded by many as the best mathematics department in the country and, while a position like that is hard to keep indefinitely, it has remained one of the strongest departments ever since. He brought in André Weil, S. S. Chern, Saunders Mac Lane and, a number of promising younger men. Moreover in the words of one of the latter “Marshall devoted himself with both intensity and breadth — from the largest issues to the smallest details — to the Department’s welfare and development”. In 1952, Stone turned the chairmanship of the Chicago department over to Saunders Mac Lane but continued to be occupied with administrative matters. He was a strong force in reestablishing the International Mathematical Union — was much involved in the drafting of its constitution and served as its president from 1952 to 1954. He also interested himself actively in the problems of mathematical and scientific teaching — especially at the international level — and served on various boards and commissions, including as president of the International Committee on Mathematical Instruction from 1961 to 1967.
While his various administrative concerns and activities prevented him from working on mathematical problems with his former intensity Stone continued to work and to publish results rather steadily until the early 1960s. In particular one should note his four papers on integration theory [15], [18], [16], [19] in which he develops an axiomatic theory of integration, building on the earlier work of Daniell, and paralleling the axiomatic theory of integration that Bourbaki was soon to publish. At the same time his mathematical interests tended more and more toward the elucidation of questions of great generality and profundity about the true nature of mathematics and mathematical concepts, for instance [15], [18], [20]. His papers [21] and [22] articulate his views on the structure and nature mathematics and his thoughts and recommendations on mathematics education, and also display his splendid command of English prose. His review [23] of Constance Reid’s biography of Courant is a delight to read. Indeed, there is reason to believe that his publications of the last thirty years give a very incomplete picture of his mathematical activity. At a conference in honor of his second retirement (see below) Stone gave a remarkable two-hour lecture outlining his rather unusual and original views on the nature and structure of mathematics.
Stone remained at the University of Chicago until he retired as Professor Emeritus in 1968. At this time there was a week-long conference in his honor the proceedings of which were published by Springer-Verlag [sic] with Felix Browder as the editor [e11]. He did not wish to stop teaching however and forthwith began a new career as George David Birkhoff professor of mathematics at the University of Massachusetts in Amherst. No doubt the fact that Amherst, Massachusetts had been his father’s childhood home added to the attractiveness of this move. He taught there for the next twelve years and among other activities supervised two Ph.D. theses. During his final year he was honored with a second retirement conference.
One of the many striking accomplishments of Marshall Stone was a truly extraordinary command of the English language. This gave a special flavor to his book, his mathematical papers and his many writings on other subjects. His skill with the English language also manifested itself in his lectures, which were models of clarity and organization.
Stone was only moderately active in supervising Ph.D. theses. Indeed there are anecdotes about his reluctance to do this sort of teaching. On the other hand, he did turn out a respectable number of new Ph.D.s and influenced many other young mathematicians by his writings and through informal personal contact. According to the Mathematics Genealogy Project, Stone had 14 doctoral students, and currently has 1455 mathematical descendants.
Stone married young (in 1927), and he and his first wife Emmy raised three daughters. Doris, Cynthia, and Phoebe. Reports have it that he was serious about fatherhood in a rather old fashioned way. His daughters had regular chores to do and in financial matters were kept on strict allowances. Like his father, Stone was a New Englander, and one of his daughters wrote that “… his social and political attitudes, his personality, all reflect his new England heritage, and one becomes aware of how deeply his roots are imbedded in this part of the world when we are together on the island (the Isle au Haut)” [e12]. On the other hand he also believed in family fun and one of his many side interests was gourmet cookery and gourmet dining. His daughters recall many delicacies that he concocted. This marriage dissolved in divorce in 1962 but Stone soon remarried. His second marriage, to Ravijojla Kostic, lasted the rest of his life. His stepdaughter Svetlana Kostic-Stone also survived him.
Of all Stone’s many interests his love of travel was surely dominant. He began to travel when he was quite young and was on a trip to India when he died. He traveled frequently and extensively and was interested in seeing all parts of the globe. For example he visited the Pacific islands and (while traveling with Ravijojla) was shipwrecked in Antartica. It is very hard to think of a place that he has not come fairly close to at some time or another. Stone, of course, was the recipient of many honors. His early election to the National Academy of Sciences in 1938 has already been noted. He was elected, also at an early age to the American Academy of Arts and Sciences in 1933 — the same year in fact that his father was also elected to the Academy [e10], and to the American Philosophical Society. He was the American Mathematical Society Colloquium Lecturer in 1939, and was honored by being selected as the Josiah Willard B. Gibbs lecturer in 1956. He served as president of the American Mathematical Society (1943–1944) and received many honorary doctorates, both domestic and foreign. In 1982 he was awarded the National Medal of Science. According to Edwin Hewitt [e12], two extra honors that gave him special pleasure were his election to an Honorary Professorship at Columbia Teachers College and to membership in the Explorers Club of New York City.
Marshall Stone was a man with a very broad outlook and wide range of interests who seems to have thought rather deeply about a number of issues. One had only to talk to him at length or read his non-mathematical writings to come away with the impression that here was an unusually thoughtful man with a high degree of penetration and insight. More than most he seemed well endowed with a quality, which one can only describe as wisdom. While on a visit to Madras, India he died quickly of a sudden illness on January 9, 1989.
