George Mackey 1916–2006 Robert S

George Mackey 1916–2006 Robert S. Doran and Arlan Ramsay

Robert S. Doran the time, were themselves fairly young Chicago Ph.D.’s. I purchased a copy of Mackey’s notes through the University of Chicago mathematics Introduction department and had them carefully bound in hard George Whitelaw Mackey died of complications cover. They became my constant companion as from pneumonia on March 15, 2006, in Belmont, I endeavored to understand this beautiful and Massachusetts, at the age of ninety. He was a re- difficult subject. In this regard it is certainly not markable individual and mathematician who made an exaggeration to say that Mackey’s notes were a lasting impact not only on the theory of infinite often the catalyst that led many mathematicians dimensional group representations, ergodic theo- to study representation theory. As one important ry, and mathematical physics but also on those example I mention J. M. G. (Michael) Fell, who, with individuals with whom he personally came into the late D. B. Lowdenslager, wrote up the original contact. The purpose of this article is to celebrate 1955 Chicago lecture notes. Fell makes it clear in and reflect on George’s life by providing a glimpse the preface of our 1988 two-volume work [1, 2] into his mathematics and his personality through on representations of locally compact groups and the eyes of several colleagues, former students, Banach *-algebraic bundles that the basic direc- family, and close friends. Each of the contributors tion of his own research was permanently altered to the article has his or her own story to tell by his experience with Mackey’s Chicago lectures. regarding how he influenced their lives and their The final chapter of the second volume treats mathematics. generalized versions of Mackey’s beautiful normal I first became aware of George Mackey forty-five subgroup analysis and is, in many respects, the years ago through his now famous 1955 Chicago apex of the entire work. It is perhaps safe to say lecture notes on group representations and also that these volumes stand largely as a tribute to through his early fundamental work on the du- George Mackey’s remarkable pioneering work on ality theory of locally convex topological vector induced representations, the imprimitivity theo- spaces. I had taken a substantial course in func- rem, and what is known today as the “Mackey tional analysis, and Mackey’s Chicago notes were Machine”. a rather natural next step resulting from my inter- In 1970 I contacted George to see if he would est in C*-algebras and von Neumann algebras—an be interested in coming to TCU (Texas Christian interest instilled in me by Henry Dye and Ster- University) to present a series of lectures on a ling Berberian, both inspiring teachers who, at topic in which he was currently interested. In spite of being very busy (at the time, as I recall, he was Robert S. Doran is Potter Professor of Mathematics at in Montecatini, Italy, giving lectures) he graciously Texas Christian University. His email address is [email protected]. accepted my invitation, and I enthusiastically sub- Arlan Ramsay is emeritus professor of mathematics at mitted a grant to the NSF for a CBMS (Conference the University of Colorado at Boulder. His email address Board of the Mathematical Sciences) conference is [email protected]. to be held at TCU with George as the principal

824 Notices of the AMS Volume 54, Number 7 lecturer. Although the grant was carefully pre- kind of mathematical argument. Perhaps this, at pared, including a detailed description provided least in part, is why he had a particularly strong by George of his ten lectures titled “Ergodic the- affinity for mathematics, a discipline where proof, ory and its significance in statistical mechanics not faith, is the order of the day. and probability theory”, the grant was not fund- Following the biography below, Calvin Moore ed. Given George’s high mathematical standing provides an overview of George’s main mathemat- and reputation, I was of course surprised when we ical contributions together with some personal were turned down. Fortunately, I was able to secure recollections. Then each of the remaining writ- funding for the conference from another source, ers shares personal and mathematical insights and in 1972 George came to TCU to take part as resulting from his or her (often close) association originally planned. True to form he showed up in with George. Arlan Ramsay and I, as organizers, Fort Worth with sport coat and tie (even though are deeply grateful to all of the writers for their it was quite hot) and with his signature clipboard contributions, and we sincerely thank them for in hand. Many of the leading mathematicians and honoring an esteemed colleague, mentor, family mathematical physicists of the day, both young member, and friend who will be greatly missed. and old, attended, and the result was a tremen- dously exciting and meaningful conference. To Biography of George W. Mackey put the cherry on top, Mackey’s ten conference George Whitelaw Mackey was born February 1, lectures were published in [3], and he received a 1916, in St. Louis, Missouri, and died on March 15, coveted AMS Steele Prize in 1975 for them. 2006, in Belmont, Massachusetts. He received a From these ear- bachelor’s degree from the Rice Institute (now Rice ly days in the University) in 1938. As an undergraduate he had 1970s George and interests in chemistry, I became good physics, and mathemat- friends, and we ics. These interests were would meet for developed during his ear- lunch on those ly high school years. occasions that we After briefly considering could get togeth- chemical engineering as a er at meetings major his freshman year and conferences (lunches are a pat- in college, he decided tern with George, instead that he wanted as you will note to be a mathematical from other writ- physicist, so he ended ers). It seemed up majoring in physics. that we agreed However, he was in- creasingly drawn to pure Photo courtesy of Ann Mackey. on most things mathematics because of George Mackey, on mathematical—he what he perceived as vacation in 1981. of course was the master, I was the a lack of mathematical student. On the other hand, he particularly liked rigor in his physics class- Photo courtesy of Ann Mackey. to debate philosophical and spiritual matters, and es. His extraordinary gift A young George Mackey. we held rather different points of view on some in mathematics became things. These differences in no way affected our particularly clear when he was recognized nation- relationship. If anything, they enhanced it. Indeed, ally as one of the top five William Lowell Putnam George honestly enjoyed pursuing ideas to the winners during his senior year at Rice. His reward very end, and he doggedly, but gently, pushed for this accomplishment was an offer of a full hard to see if one could defend a particular point scholarship to Harvard for graduate work, an offer of view. I enjoyed this too, so it usually became a that he accepted. kind of sparring match with no real winner. The He earned a master’s degree in mathematics in next time we would meet (or talk on the phone) 1939 and a Ph.D. in 1942 under the direction of it would start all over again. Although at times famed mathematician Marshall H. Stone, whose he could be somewhat brusque, he was always 1932 book Linear Transformations in Hilbert kind, compassionate, and highly respectful, and, Space had a substantial influence on his mathe- I believe, he genuinely wanted to understand the matical point of view. Through Stone’s influence, other person’s viewpoint. It seemed to me that Mackey was able to obtain a Sheldon Traveling he simply had a great deal of difficulty accepting Fellowship allowing him to split the year in 1941 statements that he personally could not establish between Cal Tech and the Institute for Advanced through careful logical reasoning or through some Study before completing his doctorate. While

August 2007 Notices of the AMS 825 at the Institute he met many legendary figures, Ph.D. Students of George Mackey, among them Albert Einstein, Oswald Veblen, and Harvard University John von Neumann, as well as a host of young Ph.D.’s such as Paul Halmos, Warren Ambrose, Lawrence Brown, 1968 Valentine Bargmann, Paul Erdos,˝ and Shizuo Paul Chernoff, 1968 Kakutani. Lawrence Corwin, 1968 After receiving his Ph.D. he spent a year on the Edward Effros, 1962 faculty at the Illinois Institute of Technology and Peter Forrest, 1972 then returned to Harvard in 1943 as an instruc- Andrew Gleason, 1950 tor in the mathematics department and remained Robert Graves, 1952 there until he retired in 1985. He became a full Peter Hahn, 1975 professor in 1956 and was appointed Landon T. Christopher Henrich, 1968 Clay Professor of Mathematics and Theoretical John Kalman, 1955 Science in 1969, a position he retained until he Adam Kleppner, 1960 retired. M. Donald MacLaren, 1962 His main areas of research were in repre- Calvin Moore, 1960 sentation theory, group actions, ergodic theory, Judith Packer, 1982 functional analysis, and mathematical physics. Richard Palais, 1956 Much of his work involved the interaction be- Arlan Ramsay, 1962 tween infinite-dimensional group representations, Caroline Series, 1976 the theory of operator algebras, and the use of Francisco Thayer, 1972 quantum logic in the mathematical foundations Seth Warner, 1955 of quantum mechanics. His notion of a system of John Wermer, 1951 imprimitivity led naturally to an analysis of the Thomas Wieting, 1973 representation theory of semidirect products in Neal Zierler, 1959 terms of ergodic actions of groups. Robert Zimmer, 1975 He served as visiting professor at many institutions, including the George Eastman professor at Oxford University; the University of Chicago; wife, Alice, typed some of his handwritten notes the University of California, Los Angeles; the Uni- and helped get the article completed and in print. versity of California, Berkeley; the Walker Ames Dick Kadison and I, as editors of the volume in professor at the University of Washington; and the which the paper appeared, are extremely grateful International Center for Theoretical Physics in Tri- for her kindness and help. este, Italy. He received the distinguished alumnus award from Rice University in 1982 and in 1985 References received a Humboldt Foundation Research Award, J. M. G. Fell R. S. Doran which he used at the Max Planck Institute in Bonn, [1] and , Representations of *-algebras, locally compact groups, and Banach Germany. *-algebraic bundles, Vol. 1 (General representation MackeywasamemberofthetheNationalAcade- theory of groups and algebras), Pure and Applied my of Sciences, the American Academy of Arts and Mathematics 125, Academic Press, New York, 1988. Sciences, and the American Philosophical Society. [2] , Representations of *-algebras, locally com- He was vice president of the American Mathemat- pact groups, and Banach *-algebraic bundles, Vol. 2 ical Society in 1964–65 and again a member of the (Banach *-algebraic bundles, induced representa- Institute for Advanced Study in 1977. tions, and the generalized Mackey analysis), Pure His published works include Mathematical and Applied Mathematics 126, Academic Press, Foundations of Quantum Mechanics (1963), Math- New York, 1988. [3] G. W. Mackey, Ergodic theory and its significance ematical Problems of Relativistic Physics (1967), in statistical mechanics and probability theory, Induced Representations of Groups and Quantum Advances in Mathematics 12 (1974), 178–268. Mechanics (1968), Theory of Unitary Group Rep- [4] , Marshall H. Stone: Mathematician, states- resentations (1976), Lectures on the Theory of man, advisor and friend, Operator Algebras, Functions of a Complex Variable (1977), Unitary Quantization, and Noncommutative Geometry, Con- Group Representations in Physics, Probability, and temporary Mathematics 365 (eds., R. S. Doran and Number Theory (1978), and numerous scholarly R. V. Kadison), Amer. Math. Soc., Providence, RI, articles. 2004, pp. 15–25. George’s final article [4], published in December of 2004, contains in-depth descriptions of some of the items mentioned in this biography as well as his interactions with Marshall Stone and others while he was a Harvard graduate student. He was not in good health at the time, and his devoted

826 Notices of the AMS Volume 54, Number 7 Calvin C. Moore classic theorem about the quantum mechanical commutation relations. He then also saw immediately that there was a version for nonabelian George Mackey’s Mathematical Work locally compact groups, which appeared in [Ma4], After graduating from the Rice Institute (as Rice but this is really part of the next phase in Mack- University was known at the time) in 1938 with ey’s work, to which we turn now. Mackey initiated a major in physics and a top five finish in the a systematic study of unitary representations of Putnam Exam, George Mackey entered Harvard general locally compact second countable (and all University for doctoral study in mathematics. He groups will be assumed to be second countable soon came under the influence of Marshall Stone without further mention) groups, work for which and in 1942 completed a dissertation under Stone he is most famous. Von Neumann had developed entitled “The Subspaces of the Conjugate of an a theory of direct integral decompositions of op- Abstract Linear Space”. In this work he explored erator algebras in the 1930s as an analog of direct the different locally convex topologies that a vec- sum decompositions for finite-dimensional alge- tor space can carry. The most significant result to bras, but he did not publish it until F. I. Mautner emerge can be stated as follows. Consider two (say persuaded him to do so in 1948. real) vector spaces V and W that are in perfect Adapted to representation theory, direct inte- duality by a pairing gral theory became a crucial tool that Mackey V × W → R used and developed. For representations of finite groups, induced representations whereby one in- so that each may be viewed as linear functionals duces a representation of a subgroup H of G up on the other. It was obvious that there is a weak- to the group G are an absolutely essential tool. est (smallest) locally convex topology on V (or Mackey in a series of papers [Ma5, Ma6, Ma7] de- W ) such that the linear functionals coming from fined and studied the process of induction of a W (or V ) are exactly the continuous ones, called unitary representation of a closed subgroup H ofa the weak topology. What Mackey proved was that locally compact group G to form the induced rep- there is a unique strongest (largest) locally convex resentation of G. When the coset space G/H has topology such that the linear functionals coming a G-invariant measure, the definition is straight- from W are the continuous ones. This is the topology of convergence of elements of V , now viewed forward, but when it has only a quasi-invariant as linear functionals on W uniformly on weakly measure, some extra work is needed. Mackey de- compact convex subsets of W . All locally convex veloped analogs of many of the main theorems topologies on V for which the linear functionals about induced representations of finite groups. from W are exactly the continuous ones lie be- In particular he established his fundamental im- tween these two [Ma1, Ma2]. This topology became primitivity theorem, which characterizes when a known generally as the Mackey topology. representation is induced. The process of induc- Mackey retained a life-long interest in theo- tion had appeared in some special cases a year or retical physics, no doubt inspired initially by his two earlier in the work of Gelfand and his collabo- undergraduate work, and soon after his initial rators on unitary representations of the classical work, he turned his attention to the theorem Lie groups. of Stone [S] and von Neumann [vN] that asserts This imprimitivity theorem states that a unitary that a family of operators p(i) and q(i) on a representation U of a group G is induced by a Hilbert space satisfying the quantum mechanical unitary representation V of a closed subgroup H commutation relations [p(k), q(j)] = iδ(k, j)I is of G if and only if there is a normal representation essentially unique. Mackey realized that it was re- of the von Neumann algebra L∞(G/H, µ) on the ally a theorem about a pair of continuous unitary same Hilbert space as U (or equivalently that there representations, one U of an abelian locally group is a projection-valued measure on G/H absolutely A, and the other V of its dual group Aˆ which continuous with respect to µ) which is covariant satisfied with respect to U in the natural sense. Here µ is a quasi-invariant measure on G/H. Moreover, U and U(s)V(t) = (s, t)V(t)U(s) V uniquely determined each other up to unitary where (s, t) is the usual pairing of the group and equivalence. Mackey’s theorem above on unicity of its dual. He showed [Ma3] that such a pair is unique pairs of representations of an abelian group A and if they jointly leave no closed subspace invariant its dual group Aˆ is a special case of applying the and in general any pair is isomorphic to a direct imprimitivity theorem to a generalized Heisenberg sum of copies of the unique irreducible pair. When group built from A and Aˆ. A is Euclidean n-space, this result becomes the These results laid the basis for what was to Calvin C. Moore is emeritus professor of mathematics at become known as the Mackey little group method, the University of California at Berkeley. His email address or as some have called it, the Mackey machine, is [email protected]. for calculating irreducible unitary representations

August 2007 Notices of the AMS 827 of a group knowing information about subgroups. absence of non-type I representations of G. Mack- But before this program could get under way, ey then made a bold conjecture that a locally Mackey had to put in place some building blocks compact group G had a smooth dual if and only or preliminaries. First some basic facts about Borel if it is type I. It was not too long before James structures needed to be laid out, which Mackey Glimm provided a proof of this conjecture [Gl]. did in [Ma9]. A Borel structure is a set together Another element was to deal with what one with a σ -field of subsets. He identified two kinds would call projective unitary representations of a of very well-behaved types of Borel structures that group G, which are continuous homomorphisms he called standard and analytic based on some from G to the projective unitary group of a Hilbert deep theorems in descriptive set theory of the space. Such projective representations not only Polish school. An equivalence relation on a Borel arise naturally in the foundations of quantum space leads to a quotient space with its own Borel mechanics, but as was clear, when one started structure. If the original space is well behaved, to analyze ordinary representations, one was led the quotient can be very nice—one of the two naturally to projective representations. By using a well-behaved types above—or if not, it is quite lifting theorem from the theory of Borel spaces, pathological. If the former holds, then the equiv- Mackey showed that a projective representation alence relation is said to be smooth. The set of could be thought of as a Borel map U from G to concrete irreducible unitary representations of a the unitary group satisfying group G can be given a natural well-behaved Borel U(s)U(t) = a(s, t)U(st), structure, and then the equivalence relation of uni- where A is a Borel function from G ×G to the circle tary equivalence yields the quotient space—that group T that satisfies a certain cocycle identity. is, the set of equivalence classes of irreducible For finite groups, it was clear that any projective unitary representations, which he termed the dual representation of a group G could be lifted to an ˆ space G of G. If the equivalence relation is well- ordinary representation of a central extension of ˆ behaved, G is a well-behaved space, and Mackey G by a cyclic group. In the locally compact case said then that G had smooth dual. This was a one would like to have the same result but with crucial concept in the program. a central extension of G by the circle group T . If Mackey developed some additional facts about the cocycle above were continuous, it would be actions of locally compact groups on Borel spaces. obvious how to construct this central extension. A group action ergodic with respect to a quasi- Mackey, by a very clever use of Weil’s theorem invariant measure—and ergodicity is a concept on the converse to Haar measure, showed how to that played such a central role in Mackey’s work construct this central extension. He also began an over decades—is one whose only fixed points in exploration of some aspects of the cohomology the measure algebra are 0 and 1. An important theory that lay in the background [Ma8]. observation is that if the equivalence relation in- Mackey’s little group method starts with a group duced on X by the action of G is smooth, then G for which we want to compute Gˆ, and it is as- any ergodic measure is concentrated on an or- sumed that N is a closed normal subgroup that bit of G, and so up to null sets the action is has a smooth dual (and is hence now known to be transitive. Another related issue concerned point type I). It is assumed that Nˆ is known, and then G realizations of actions of a group. Suppose that (or really G/N) acts on Nˆ as a Borel transformation G acts as a continuous transformation group on group. Any irreducible representation U of G yields the measure algebra M(X, µ) of a measure space, upon restriction to N a direct integral decomposi- where X is a well-behaved Borel space. Then, can tion into multiples of irreducible representations one modify X by µ-null sets if necessary and show with respect to a measure µ on Nˆ, which Mackey that this action comes from a Borel action of G on showed was ergodic. Then if the quotient space of the underlying space X that leaves the measure Nˆ by this action is smooth, any ergodic measure is quasi-invariant? In [Ma12] Mackey showed that the transitive and is carried on some orbit G · V of G answer is affirmative, extending an earlier result on Nˆ. Hence the representation U has a transitive of von Neumann for actions of the real line. system of imprimitivity based on G/H where H is Also, by adapting von Neumann’s type theory the isotropy group of V . Hence U is induced by for operator algebras, Mackey introduced the no- a unique irreducible representation of H, whose tion of a type I group, by which he meant that all restriction to N is a multiple of V , and these can its representations were type I or equivalently all be classified in terms of irreducible representa- of its primary representations were multiples of tions or projective representations of H/N, which an irreducible representation. On the basis of his is called the “little group”. In an article in 1958 work in classifying irreducible representations of a Mackey laid out his systematic theory of projective group—e.g., calculating Gˆ—Mackey observed that representations [Ma11]. Mackey’s work also builds the property of a group G having a smooth dual on Wigner’s analysis in 1939 of the special case seemed to be related to and correlated with the of unitary representations of the inhomogeneous

828 Notices of the AMS Volume 54, Number 7 Lorentz group [W]. Mackey’s little group method, in subsequent work to make it function properly. an enormously effective, systematic tool for ana- In a real sense, the point of view introduced here lyzing representations of many different groups, by Mackey was the opening shot in the whole pro- was used to good effect by many workers, and has gram of noncommutative topology and geometry been extended in different directions. that was to develop. One particularly rich theme In the summer of 1955 Mackey was invited has emerged from the special case when the group to be a visiting professor at the University of action is free, in which case the groupoid is simply Chicago, and he gave a course that laid out his an equivalence relation, and Mackey defines what theory of group representations that we have one means by a measured ergodic equivalence briefly described. Notes from the course prepared relation. Isomorphism of measured equivalence by J. M. G. Fell and D. B. Lowenslager circulated relations amounts to orbit equivalence of the informally for years, and generations of students group actions, a notion that was foreign in ergodic (including the author) learned Mackey’s theory theory up to that point but which has been of over- from these famous notes. In 1976 Mackey agreed riding importance in developments since then. In to publish an edited version of these notes, to- fact [Ma14] adumbrates some of the subsequent gether with an expository article summarizing developments that have sprung from his work. progress in the field since 1955 [Ma16]. Of course As has already the Mackey machine runs into trouble when the been suggested, action of G on Nˆ is not smooth, and nontransitive Mackey main- ergodic measures appear in the analysis above. In tained a live- his 1961 AMS Colloquium Lectures [Ma13] Mack- ly and inquiring ey laid out a new approach to this problem and lifelong interest introduced the notion of virtual group. He ob- in mathemati- served that a Borel group action of a group G on cal physics and a measure space Y defines a groupoid—a set with especially the a partially defined multiplication where inverses foundations of exist. The groupoid consists of Y × G where the quantum theo- product (y, g) · (z, h) is defined when z = y · g, ry, quantum field and the product is (y, gh) where it is convenient theory, and sta- to write G as operating on the right. This set has tistical mechan- a Borel structure and a measure—the measure µ ics. In [Ma10] on Y cross Haar measure that has the appropri- Mackey explored ate “invariance” properties. If the measure µ is the abstract rela- ergodic, then Mackey called the construction an tionship between Photo courtesy of Ann Mackey. ergodic measured groupoid. Mackey realized that quantum states different such objects needed to be grouped to- and quantum ob- Mackey at the University of Chicago, gether under a notion he called similarity, and he servables and 1955. defined a virtual group to be an equivalence class raised the question of whether some very gen- under similarity of ergodic measure groupoids. In eral axioms about that relationship necessarily led the case of a groupoid coming from a transitive to the classical von Neumann formulation. This action of G on a coset space H\G of itself, the exposition inspired Andrew Gleason to prove a similarity notion makes the measured groupoid strengthened version of Mackey’s results, which H\G × G similar to the measured groupoid that is then enabled him to formulate a general result simply the group H (with Haar measure) and so that showed that the von Neumann formulation puts them in the same equivalence class. Hence followed from a much weaker set of axioms. Also in this case the transitive measured groupoid is Mackey’s work on the unicity of the Heisenberg literally a subgroup of G, and Mackey’s point here commutation relations gave an indication why, is that it would be very productive to look at a when the number of p(i)’s and q(i)’s is infinite general ergodic action of G as a kind of general- (quantum field theory), the uniqueness breaks ized (or virtual) subgroup of G via the language down. of groupoids and virtual groups. Then one could In his later years Mackey wrote a number of begin a systematic study of the representations of fascinating historical and integrative papers and virtual groups, induced representations, etc. The books about group representations, and harmonic imprimitivity theorem remains true in the ergodic analysis and its applications and significance for nontransitive case in that the irreducible repre- other areas of mathematics and in the mathemat- sentation of G is now induced by an irreducible ical foundations of physics. The theme of Norbert representation of a virtual subgroup. Wiener’s definition of chaos, or homogeneous Mackey laid out his theory in two subsequent chaos, was a favorite theme in his writings. Also publications in 1963 and 1966 [Ma14, Ma15]. His applications of group representations to number initial notion of similarity had to be adjusted a bit theory was another common theme, among many

August 2007 Notices of the AMS 829 others. Mackey was invited to be the George East- MSRI found him and Alice a beautiful rental house man Visiting Professor at Oxford University for for the year that belonged to Geoff Chew, a faculty 1966–67, and he and Alice spent the year in Oxford. member in physics who was on sabbatical for the He gave a broad-ranging series of lectures dur- year. The only problem was that the house came ing the year, which he subsequently published in with some animals, cats and dogs, that the tenants 1978 as Unitary Group Representations in Physics, would have to take care of. But the Mackeys said Probability, and Number Theory [Ma17]. George that would be no problem. George arrived a month Mackey will be remembered and honored for his or two before Alice could come so that the house seminal contributions to group representations would not be vacant with no one to take care of the and ergodic theory and mathematical physics and animals. George truly had his hands full with the for his fascinating expositions on these subjects. animals, and even after Alice arrived to take over Let me close this summary of Mackey’s research housekeeping, they had many amusing incidents. contributions with some personal thoughts. I first When they returned to Cambridge after their year met George Mackey in 1956 when I was entering in the “Wild West”, Alice wrote a fascinating and my junior year at Harvard. He became my advi- hilarious article for the Wellesley alumni maga- sor and mentor both as an undergraduate and zine about George’s and her travails with the Chew when I continued as a graduate student at Har- menagerie. vard. I learned how to be a mathematician from him, and I valued his friendship, guidance, and References encouragement for over fifty years. Many of my [Gl] James Glimm, Type I C∗-Algebras, Ann. Math. own accomplishments can be traced back to ideas 73 (1961), 572–612. and inspirations coming from him. George was a [Ma1] George W. Mackey, On convex topological lin- uniquely gifted and inspiring individual, and we ear spaces, Proc. Nat. Acad. Sci. U.S.A. 29 (1943), 315–319, and Erratum 30, 24. miss him very much. [Ma2] , On convex topological linear spaces, Trans. Amer. Math. Soc. 60 (1946), 519–537. [Ma3] , A theorem of Stone and von Neumann, Duke Math. J. 16 (1949), 313–326. [Ma4] , Imprimitivity for representations of locally compact groups, I. Proc. Nat. Acad. Sci. U.S.A. 35 (1949), 537–545. [Ma5] , On induced representations of groups, Amer. J. Math. 73 (1951), 576–592. [Ma6] , Induced representations of locally compact groups, I, Ann. of Math. (2) 55 (1952), 101–139. [Ma7] , Induced representations of locally compact groups, II, The Frobenius reciprocity theorem, Ann. of Math. (2) 58 (1953), 193–221. [Ma8] , Les ensembles boréliens et les exten- Photo courtesy of Arthur Jaffe. sions des groupes, J. Math. Pures Appl. (9) 36 George and Alice Mackey, May 1984, Berkeley (1957), 171–178. Faculty Club, at a conference in honor of [Ma9] , Borel structure in groups and their Mackey. duals, Trans. Amer. Math. Soc. 85 (1957), 134–165. George visited Berkeley on several occasions, [Ma10] , Quantum mechanics and Hilbert space, and two incidents stick in my mind that are in Amer. Math. Monthly 64 (1957), no. 8, part II, 45–57. some ways characteristic of him. One time, prob- [Ma11] , Unitary representations of group exten- ably in the 1960s or 1970s when George was sions, I, Acta Math. 99 (1958), 265–311. visiting Berkeley, a group of us were walking to [Ma12] , Point realizations of transformation lunch and talking mathematics. In this discussion groups, Illinois J. Math. 6 (1962), 327–335. I described a certain theorem that was relevant for [Ma13] , Infinite-dimensional group represen- the discussion (unfortunately I cannot recall what tations, Bull. Amer. Math. Soc. 69 (1963), the theorem was), and George remarked to the 628–686. effect, “Well, that’s a very nice result. Who proved [Ma14] , Ergodic theory, group theory, and differ- it?” My response was “You proved it.” Well, from ential geometry, Proc. Nat. Acad. Sci. U.S.A. 50 one perspective it is nice to have proved so many (1963), 1184–1191. [Ma15] , Ergodic theory and virtual groups, good theorems that you can forget a few. Math. Ann. 166 (1966), 187–207. The other incident or series of incidents oc- [Ma16] , The Theory of Unitary Group Represen- curred in 1983–84, when I had arranged for George tations, based on notes by James M. G. Fell and to visit at the Mathematical Sciences Research In- David B. Lowdenslager of lectures given at the stitute (MSRI) for a year. The housing officer at University of Chicago, Chicago, IL, 1955, Chicago

830 Notices of the AMS Volume 54, Number 7 Lectures in Mathematics, University of Chicago idea, though, in hindsight, I think I conceived of Press, Chicago, IL/London, 1976, x+372 pp. it in a naive and narrow manner. But it appealed [Ma17] , Unitary Group Representations in to me because of the beauty of the concept of Physics, Probability, and Number Theory, symmetry as a fundamental fact of nature. Never Mathematics Lecture Note Series, 55, Ben- mind whether it was in conformity with recent jamin/Cummings Publishing Co., Inc., Reading, discoveries in physics!! MA, 1978, xiv+402 pp. [vN] John von Neumann, Die eindeutigkeit der But George’s mind was much broader than this. Schrödingerschen operatoren, Math. Ann. 104 He seemed willing to embrace scientific reality (1931), 570–578. wherever he found it. One of the aspects of his [S] M. H. Stone, Linear transformations in Hilbert mathematical creativity that especially struck me space III, operational methods and group theory, was his interest—and success—in finding appli- Proc. Nat. Acad. Sci. U.S.A. 16 (1930), 172–175. cations of the theory of group representations [W] Eugene Wigner, Unitary representations of the to quite different fields of mathematics, includ- inhomogeneous Lorentz group, Ann. Math. 40 ing mathematical physics! It seems that his mind (1939), 149–204. was impelling him toward the ideal of a universal mathematician, so hard to attain these days of ever-increasing specialization. J. M. G. (Michael) Fell Moreover, he had an idealistic, one might say “aristocratic”, view of what it takes to be a true Recollections of George Mackey mathematician. I remember him saying once that there are two kinds of mathematicians (and I sup- If there was one individual who influenced the pose that he would have made the same distinction course of my mathematical life more than any for any field of intellectual endeavor): there are other, it was George Mackey. I first met George those who are “inner-directed” who work because (about four years after receiving my Ph.D.) at the they are impelled by something within them, and 1955 Summer Conference on Functional Analysis there are those who are “outer-directed” who are and Group Representations, held at the Universi- content to have their work directed for them by ty of Chicago. George’s lectures there on group the force of outward circumstances. There is no representations were an inspiration to me. His doubt which of these two classes George Mackey contagious enthusiasm spurred me to join with was able to belong to. David Lowenslager in writing up the notes of his Unfortunately I had little or no personal con- lectures and aroused in me an enthusiasm for tact with George after about 1970. But in the years his approach to the subject which has lasted all when we knew him personally, my wife and I through my mathematical life. always found George—and his good wife, Alice, I would like to make a few remarks about why also—to be very friendly and approachable. Along I found George’s approach to group representa- with his absorption in mathematics, he had an tions so appealing (though I must apologize here interest in other people. He also had a dry though for the fact that these remarks say more about ready sense of humor. I remember hearing of one me than about George!). His approach fascinated episode in the Harvard mathematics department me because it seemed to have a beauty and uni- when a young secretary was telling this distin- versality that were almost Pythagorean in scope. guished professor what he ought to do in a rather We live in a universe whose laws are invariant bossy manner. George’s reply to her was simply under a certain symmetry group (for example, “You’re talking to me just as if you were my the Lorentz group in the case of the universe mother.” of special relativity). It seems plausible that the Now that George is no longer with us, I am one kinds of “irreducible” particles that can exist in a of those who regret not taking the opportunity to quantum-mechanical universe should be correlat- meet him and his family during his later years. And ed with the possible irreducible representations my wife and I want to join with all his other friends of its underlying symmetry group. If this is so, in condolences to his family over the departure of then it should be a physically meaningful project a great man from our midst. to classify all the possible irreducible representations of that group. And now here in George’s lectures was a three-phase program laid out as a first step toward just that purpose—indeed, for classifying the irreducible representations of all possible symmetry groups! To my mind this was an extremely exciting and emotionally satisfying

J. M. G. Fell is emeritus professor of mathematics at the University of Pennsylvania, Philadelphia.

August 2007 Notices of the AMS 831 Roger Howe They were not at all what I expected. I had been attracted to representation theory by neat for- Recollections of George Mackey mulas: Schur orthogonality relations, Bessel functions, convolution products. But George’s notes George Mackey was my mathematical grandfather. were full of Borel sets and Polish spaces and That is, he was the advisor of my advisor, Calvin projection-valued measures. They were hard to Moore of the University of California at Berkeley. understand, had a lot of unproved assertions, and According to the Mathematics Genealogy Project, were all in all quite strange. Also, they were long! this is a mathematical lineage that goes back to I despaired of making any sense of them at all in Euler and Leibniz through Marshall Stone, G. D. the short time I had. Birkhoff, E. H. Moore, and Simeon Poisson. The Ge- I studied what I could, and George did pass me. nealogy Project lists nearly 40,000 mathematical He was known for his directness, and his remarks descendants for Euler, of which about 300 come at the end of the exam were quite representative. from Mackey. The genealogy lists are updated He didn’t say I had done well. He said that I had regularly, so these numbers will increase in the learned “enough” and that I had done “better than future. Also, the Genealogy Project now lists me he expected”. as a mathematical grandfather, so George died at In retrospect, I think three things saved me: least a mathematical great great-grandfather. 1) The exam was postponed. This was actually Although George was mathematically speaking not on my account; George discovered conflicts my grandfather, his influence on me was direct, and moved the date. not just through Calvin Moore. In fact, I metGeorge 2) There were no proofs to learn. George was before I met Cal. This is something you can’t do not particularly interested in the details of proofs, in ordinary genealogy. but much more concerned with the overall struc- I took a course from George in my senior year ture of the subject. It was easier to get the gist of at Harvard. I had gotten interested in harmonic this than to put together the details of an intricate analysis and representation theory, and everyone proof. spokeof ProfessorMackey(as we called him) as the 3) George really was a wonderful expositor. I local guru on these subjects. In my last semester, didn’t understand this at the time because of the he was giving a course on representation theory. I newness and strangeness of the material, but since very much wanted to take it, but it would not fit then I have read with pleasure and benefit many of in my schedule. When I spoke to him about the his expositions—of quantum mechanics, induced problem, he volunteered that he was planning to representations, applications of and history of produce detailed written notes and I could take it representation theory. He had a marvelous talent as a reading course, so that’s what I did. for combining simple ideas to construct rich and The last coherent pictures of broad areas of mathematics. semester of I continue to recommend his works to younger senior year scholars. is a time I am grateful that that exam was not the last when one’s time I saw George. In fact, our paths crossed reg- attention is ularly, and we developed a cordial relationship. often not The next time I saw him after Harvard was at a on studies. conference at the Battelle Institute in Seattle in I turned in the summer of 1969. It was my first professional a five-page conference and one of the most delightful I have paper for ever attended. It was intended to encourage in- anarthisto- teraction between mathematicians and physicists ry course a around the applications of representation theory month late. to physics, so there were both mathematicians Also, I had Photo courtesy of Ann Mackey. and physicists there and a sense of addressing to write my important issues. Using representation theory in George Mackey (left) with his Ph.D. advisor senior the- physics was perhaps George’s favorite topic of Marshall Stone, 1984. sis. That was my first encounter with mathematics re- thought and conversation. I have memories of search, and it was very unsettling—somewhat like basking in the sun around the Battelle Institute being possessed. It cost me a lot of sleep. Anyway, grounds or on various excursions and in the con- it was late in the semester before I even looked at versations of the senior scholars: George, in his the course notes. signature seersucker pinstripe jacket; Cal Moore; B. Kostant; V. Bargmann; and others. Roger Howe is professor of mathematics at Yale Over the years since, I enjoyed many conver- University. His email address is [email protected]. sations with George, sometimes in groups and

832 Notices of the AMS Volume 54, Number 7 occasionally one-on-one. I have always been im- them on: my latest student, Hadi Salmasian, has pressed by the independence of his views—he used these same ideas to take the line of work came to his own conclusions and advanced them further and show that what had seemed perhaps with conviction born of long thought—and by his ad hoc constructions for classical groups could scholarship—he carefully studied relevant papers be seen as a natural part of the representation in mathematics or physics and took them in- theory of any semisimple or reductive group. to account, sometimes accepting, sometimes not, George’s body may have given up the ghost, but according to what seemed right. I particularly his spirit and his mathematics will be with us for remember a short but highly referenced oral dis- a long time to come. sertation on the Higgs Boson, delivered for my sole benefit. I forget when George took to referring to me as Arthur Jaffe his grandstudent, but a particularly memorable occasion when he did so was in introducing me to his Lunch with George advisor, Marshall Stone. The Stone-von Neumann Theorem, which originated as a mathematical Background characterization of the Heisenberg canonical commutation relations, was reinterpreted by Mackey I was delighted to see that the program of the as a classification theorem for the unitary repre- 2007 New Orleans AMS meeting listed me cor- sentations of certain nilpotent groups. Both Cal rectly as a student; in fact I have been a student Moore and I have found new interpretations and of George Mackey practically all my mathemati- applications for it, and now my students use it cal life. George loved interesting and provoking in their work. In fall 2004 I attended a program mathematical conversations, and we had many at the Newton Institute on quantum informa- over lunch, explaining my congenial title. tion theory (QIT). There I learned that QIT had Most of our spurred new interest in Hilbert space geometry. individual meet- One topic that had attracted substantial atten- ings began at tion was mutually unbiased bases. Two bases {uj } the Harvard Fac- and {vj }, 1 ≤ j ≤ dim H, of a finite-dimensional ulty Club. George Hilbert space H, are called mutually unbiased if walked there from the inner product of uj with vk has absolute val- working at home 1 1 2 to meet for our ue dim H , independent of j and k. A number of luncheon, and I of- constructions of such bases had been given, and ten watched him some relations to group theory had been found. pass the reading The topic attracted me, and in thinking about it,

I was amazed and delighted to see that George’s room windows. Photo courtesy of Ann Mackey. Generally our con- work on induced representations, systems of im- Mackey in Harvard office. primitivity, and the Heisenberg group combined versations engaged to give a natural and highly effective theory and us so we continued construction of large families of mutually unbi- afterward in one of our offices, which for years ased bases. It seemed quite wonderful that ideas adjoined each other in the mathematics library. that George had introduced to clarify the founda- Some other occasions also provided opportunity tions of quantum mechanics would have such a for conversation: thirty years ago the department satisfying application to this very different aspect met over lunch at the Faculty Club. Frequent- of the subject. I presented my preprint on the sub- ly we also exchanged invitations for dinner at ject to George, but at that time his health was in each other’s home. Both customs had declined decline, and I am afraid he was not able to share significantly in recent years. Another central fix- my pleasure at this unexpected application. ture revolved about the mathematics colloquium, I hadn’t expected the strange-seeming ideas which for years George organized at Harvard. in George’s notes for that reading course to im- George and Lars Alfhors invariably attended the pinge on my research. I had quite different, more dinner, and for many years a party followed in algebraic and geometric, ideas about how to ap- someone’s home. George also made sure that proach representation theory. But impinge they each participant paid their exact share of the bill, did. When I was struggling to understand some a role that could not mask the generous side of qualitative properties of unitary representations his character. of classical Lie groups, I found that the ideas from that course were exactly what I needed. And I am Arthur Jaffe is the Landon T. Clay Professor of Mathe- extremely happy not only to have used them (and matics and Theoretical Science at Harvard University. His to have had them to use!) but also to have passed email address is [email protected].

August 2007 Notices of the AMS 833 George also enjoyed lunch at the “long table” ago, the theme “The Mathematical Theory of Ele- in the Faculty Club, where a group of regulars mentary Particles” represented more dream than gathered weekly. Occasionally I joined him there reality. or more recently at the American Academy of I knew George’s excellent book on the math- Arts and Sciences, near the Harvard campus. I ematical foundations of quantum theory, so I could count on meeting George at those places looked forward to meeting him and to discussing without planning in advance. Through these in- the laws of particle physics and quantum field teractions my informal teacher became one of my theory. George was forty-nine, and I was still a best Harvard friends. So it was natural that our student at Princeton. Perhaps the youngest per- conversations ultimately led to pleasant evenings son at the meeting, I arrived in awe among many at 25 Coolidge Hill Road, where Alice and George experts whose work I had come to admire. George were gracious and generous hosts, and on other and I enjoyed a number of interesting interac- occasions to 27 Lancaster Street. tions on that occasion, including our first lunch While the main topic of our luncheons focused together. on mathematics, it was usual that the topic of Our paths crossed again two years later, on- conversation veered to a variety of other subjects, ly weeks before my moving from Stanford to including social questions of the time and even Harvard. That summer we both attended the to novels by David Lodge or Anthony Trollope. “Rochester Conference”, which brought together George seemed to come up with a viewpoint on particle physicists every couple of years. Return- any topic somewhat orthogonal to mine or to ing in 1967 to the University of Rochester where other companions, but one that he defended both the series began, the organizers made an attempt with glee as well as success. to involve some mathematicians as well. George began as a student of physics and The Rochester hosts prepared the proceedings found ideas in physics central to his mathemat- in style. Not only do they include the lectures, but ics. Yet George could be called a “quantum field they also include transcripts of the extemporane- theory skeptic”. He never worked directly on this ous discussions afterward. Today those informal subject, and he remained unsure whether quan- interchanges remain of interest, providing far tum mechanics could be shown to be compatible better insights into the thinking of the time than with special relativity in the framework of the the prepared lectures that precede them. The Wightman (or any other) axioms for quantum discussion following the lecture by Arthur Wight- fields. man includes comments by George Mackey, Irving When we began to interact, the possibility to Segal, Klaus Hepp, Rudolf Haag, Stanley Mandel- give a mathematical foundation to any complete stam, Eugene Wigner, C.-N. Yang, and Richard example of a relativistic, nonlinear quantum field Feynman. It is hard to imagine that diverse a appeared far beyond reach. Yet during the first ten years of our acquaintance these mathemat- spectrum of scientists, from mathematicians to ical questions underwent a dramatic transition, physicists, sitting in the same lecture hall—much and the first examples fell into place. George less discussing a lecture among themselves! and I discussed this work many times, reviewing Reading the text with hindsight, I am struck by how models of quantum field theory in two- and how the remarks of Mackey and of Feynman hit three-dimensional Minkowski space-time could be the bull’s-eye. George’s comments from the point achieved. While this problem still remains open in of view of ergodic theory apply to the physical four dimensions, our understanding and intuition picture of the vacuum. Feynman’s attitude about have advanced to the point that suggests one may mathematics has been characterized by “It is a find a positive answer for Yang-Mills theory. Yet theorem that a mathematician cannot prove a George remained unsure about whether this cul- nontrivial theorem, as every proved theorem is mination of the program is possible, rightfully trivial,” in Surely You’re Joking, Mr. Feynman. questioning whether a more sophisticated con- Yet in Rochester, Feynman was intent to know cept of space-time would revolutionize our view whether quantum electrodynamics could be (or of physics. had been) put on a solid mathematical footing. Despite this skepticism, George’s deep in- Today we think it unlikely, unlike the situation sights, especially those in ergodic theory, con- for Yang-Mills theory. nected in uncanny ways to the ongoing progress in quantum field theory throughout his lifetime. Harvard George chaired the mathematics department Early Encounters when I arrived at Harvard in 1967, and from I first met George face-to-face during a conference that time we saw each other frequently. We had organized in September 1965 by Irving Segal and our private meetings, and we each represented Roe Goodman at Endicott House. Some 41+ years our departments on the Committee for Applied

834 Notices of the AMS Volume 54, Number 7 Mathematics, yet another opportunity to lunch voting member while still retaining my original together. affiliation with physics. At that point I began During 1968, Jim Glimm and I gave the first to interact with George even more. Following mathematical proof of the existence of the unitary George’s retirement in 1985 as the first occupant group generated by a Hamiltonian for a nonlinear of his named mathematics professorship, I was quantum field in two dimensions. This was a humbled to be appointed as the successor to problem with a long history. George’s old and George’s chair. I knew that these were huge shoes dear friend Irving Segal had studied this question to fill. for years, and he became upset when he learned of its solution. Government At a lunch during April 1969 George asked George often gave advice. While this advice might me my opinion about “the letter”, to which I re- appear at first to be off-the-mark, George could sponded, “What letter?” George was referring to defend its veracity with eloquence. And only after an eight-page letter from Segal addressed to Jim time did the truth of his predictions emerge. One and me but which neither of us had received at topic dominated all others about science policy: the time. The letter claimed to point out, among George distrusted the role of government fund- other things, potential gaps in the logic of our ing. published self-adjointness proof. On finally re- George often expressed interest in the fact that ceiving a copy of the letter from the author, I had a government research grant. I did this in I realized immediately that his points did not order to be able to assist students and to hire represent gaps in logic, but they would require extraordinary persons interested in collaboration. a time-consuming response. I spent consider- George often explained why he believed scientists able effort over the next two weeks to prepare a should avoid taking government research money. careful and detailed answer. His theory was simple: the funder over time will This put George in a difficult position, but his ultimately direct the worker and perhaps play a reaction was typical: George decided to get to role out of proportion. the bottom of the mess. This attitude not only When the government funding of research reflected George’s extreme curiosity but also his evolved in the 1950s, it seemed at first to work tendency to help a friend in need. It meant too reasonably well. It certainly fueled the expan- that George had to invest considerable time and sion of university science in this country during energy to understand the details of a subtle proof the 1960s and the early 1970s. At that time I somewhat outside his main area of expertise. And believed that the government agencies did a rea- for that effort I am extraordinarily grateful. sonable job in shepherding and nurturing science. It took George weeks to wade through the pub- The scheme attempted to identify talented and lished paper and the correspondence. Although productive researchers and to assist those per- he did ask a few technical questions along the sons in whatever directions their research drew way, George loved to work things out himself at them. This support represented a subsidy for the his own pace. Ultimately George announced (over universities. lunch) the result of his efforts: he had told his But over time one saw an evolution in the old friend Segal that in his opinion the published 1970s, much in the way that George had warned. proof of his younger colleagues was correct. This settled the matter in George’s mind once and for Today the universities have became completely all. dependent on government support. On the other We returned to this theme in the summer of hand, the government agencies take the initiative 1970 when George, Alice, and their daughter, to direct and to micro-manage the direction of sci- Ann, spent two long but wonderful months at a ence, funneling money to programs that appear marathon summer school in Les Houches, over- fashionable or “in the national interest”. George looking the French Alps. George (as well as R. Bott warned that such an evolution could undermine and A. Andreotti) were observers for the Battelle the academic independence of the universities, Institute, who sponsored the school. During two as well as their academic excellence and intellec- weeks I gave fifteen hours of lectures on the orig- tual standards. It could have a devastating effect inal work and on later developments—perhaps on American science as a whole. While we have the most taxing course I ever gave. That summer I moved far in the direction of emphasizing pro- got to know the Mackeys well, as the participants grams over discovering and empowering talent, dined together almost every day over those eight one wonders whether one can alter the apparent weeks. asymptotic state. Gradually my research and publications became more and more centered in mathematics Personal Matters than physics, and in 1973 the mathematics de- George spoke often about the need to use valu- partment at Harvard invited me to become a full able time as well as possible. And the most

August 2007 Notices of the AMS 835 important point was to conserve productive time special person abound throughout mathematics. for work. Like me, George had his best ideas But they also can be heard over lunch at the early in the morning. I was unmarried when our long table in the Faculty Club and at the weekly discussions began, and George emphasized to me luncheons at the American Academy. I am not the need to have a very clear understanding with alone. Everyone misses our fascinating luncheon a partner about keeping working time sacrosanct. companion and friend.

David Mumford

To George, My Friend and Teacher* As a mathematician who worked first in algebraic geometry and later on mathematical models of perception, my research did not overlap very much with George’s. But he was, nonetheless, one of the biggest influences on my mathematical career and a very close friend. I met George in the fall of 1954—fifty-three years ago. I was a sophomore at Harvard and was assigned to Kirkland House, known then as a jock house. In this unlikely place, George was a nonresident tutor, and

Photo courtesy of Arthur Jaffe. we began to meet weekly for lunch. My father had Ushers at the wedding of Arthur Jaffe, died three years earlier, and, my being a confused September 1992, (left to right): Raoul Bott, and precocious kid, George became a second fa- Bernard Saint Donat, George Mackey, Arthur ther to me. Not that we talked about life! No, he Jaffe, James Glimm, Konrad Osterwalder. showed me what a beautiful world mathematics is. We worked through his lecture notes, and I ate them up. He showed me the internal logic George also described at length how he enjoyed and coherence of mathematics. It was his person- his close relationship with Alice and how they en- al version of the Bourbaki vision, one in which joyed many joint private activities, including read- groups played the central role. Topological vector ing novels to each other, entertaining friends and spaces, operator theory, Lie groups, and group relatives, and traveling. He also described how he representations were the core, but it was also the even limited time with daughter Ann. But when lucid sequence of definitions and theorems that he was with Ann, he devoted his total attention to was so enticing—a yellow-brick road to more and her to the exclusion of all else. more amazing places. George floated multiple warnings about mar- This was my first exposure to what higher riage that I undoubtedly should have taken more mathematics is all about. I had other mentors— seriously. But years later when I remarried, George Oscar Zariski, who radiated the mystery of math- served as an usher on that occasion; he even end- ematics; Grothendieck, who simply flew—but ed up driving the minister to the wedding in the George opened the doors and welcomed me into countryside. Afterwards George shared a surpris- the fold. In those days he led the life of an English ing thought: my wedding was the first wedding don, living in a small apartment with one arm- that he thoroughly enjoyed! In honor of that con- chair and a stereo. Here was another side of the vivial bond, I wore the necktie chosen for me and life of the intellectual: total devotion to your field, the ushers at my wedding at my presentation in which was something I had never encountered so the Special Session for George in New Orleans. intensely in anyone in my family’s circle. When Shortly after George retired, I served as depart- I graduated, my mother came to Cambridge and ment chair. At the beginning of my term I made wanted to meet one of my professors. We had a strong case that the department needed more lunch with George. After that, she said, “This is office space, as several members had no regular what I always thought a Harvard professor would office. Within a year we were able to construct be like, the real thing”. seventeen new offices in contiguous space that Back in the 1960s, government funding of had been used for storage and equipment. But mathematical research was just starting, so of before that happened, I had to ask George if he would move from his large office of many years David Mumford is University Professor in the Division of to a smaller one next door. As usual, George Applied Mathematics at Brown University. His email ad- understood and graciously obliged. dress is [email protected]. George’s straightforward analysis of the world *This note is adapted from David Mumford’s address at left one completely disarmed. Memories of this G. Mackey’s memorial.

836 Notices of the AMS Volume 54, Number 7 course everyone was applying. Not George. He physics. He devoted many years to making mani- rocked the Boston mathematical community—not fest the links between mathematics and physics. for the last time—by saying what no one else Many mathematicians have been frustrated by the dared: the government was wasting its money, seeming intractability of the problem of reducing because all of us would do math all year without quantum field theory to precise mathematics. But the two-ninths raise they were offering. He would here George was the perennial optimist: for the not take it. Besides, on a darker note, he predicted whole of life he remained sure that the ultimate all too accurately that when we were bought, the synthesis was around the corner, and he never government would try to influence our research. dropped this quest. As I learned many years To varying degrees in different fields, this has ago from reading about them in his notes, inter- come to pass. As an applied mathematician for twining operators were one of George’s favorite the last twenty years, it is downright embarrass- mathematical constructs. But I think they are a ing to see how much government pressure is metaphor for much more in George’s life. His wife being applied to create interdisciplinary collabo- and daughter were his wonderful support system, rations. If they happen, great. But this should be and they intertwined George with the real world. each mathematician’s personal choice, governed There were many wonderful gatherings at their by his interests. house. George and Alice maintained the long tra- George’s outspokenness and his brutal honesty dition of proper and gracious dinner parties for probably got under everyone’s skin at some point. the mathematical community in Cambridge, long He never adjusted his message to his listener. But after it had gone out of fashion for the younger he often articulated thoughts that we shied away generations. His family was truly his lifeline, the from. Certainly, his carrying his clipboard and hose bringing air to that diving suit. catching an hour to do math alone while his wife and daughter went to a museum is a fantasy many mathematicians harbor. George maintained Judith A. Packer his intellectual schedule through thick and thin. Perhaps my favorite memory of when he voiced a George Mackey: A Personal Remembrance totally unorthodox point of view is this. I asked him once how he survived his three years of George, or Professor Mackey, as I addressed him Harvard’s relentlessly rotating chairmanship. His during my graduate school years, made an in- reply: he was most proud of the fact that, under delible impression on me and changed my life his watch, nothing had changed; he left every- for the better, several times: firstly as a brilliant thing just as he had found it. For him, a true mathematician and my thesis advisor, secondly conservative, the right values never changed. as a mentor, and finally, as a very kind and sym- As I said, we were close friends for all his life. pathetic friend who seemed always to give good In fact, we continued to meet for lunch, George’s advice. favorite way of keeping in touch, until his deteri- He first entered my radar screen in the fall of orating health overtook him and he was forced to 1978, just after I had entered Harvard Universi- retreat to a nursing home. He would always walk ty’s mathematics department graduate program. from his house on Coolidge Hill to the Faculty My undergraduate advisor, Ethan Coven, had sug- Club—perhaps this was his chief source of phys- gested that I talk with Mackey, since I had just ical exercise. Over lunch, we would first go over finished a senior thesis on symbolic dynamics what kind of math each of us was playing with and I thought I knew some ergodic theory. “You at that time. But then we also talked philosophy must meet George Mackey; he works in ergodic and history, both of which attracted George a theory,” said Ethan, “but you might not recognize great deal. He liked the idea that perhaps, as it from what you have been studying in your conscious beings, we might not really be living in senior thesis. He has a broader approach.” This this world. He used the metaphor, for instance turned out to be an understatement of some large (like the movie The Matrix), that our real body order of magnitude. At the first tea I attended that could be elsewhere but wearing a diving suit that fall, I immediately spotted George. He was hard reproduced the sensations and transmitted our to miss, standing in the center of the tea room in motions to the object that we usually conceive of his tweed jacket, holding forth on some topic of as our body. The conventional reality of our lives import and being gregarious to all who came his might be a pure illusion. way, especially if they wanted to talk to him about George was not religious in the conventional mathematics or mathematical personalities. I met sense. He would certainly reject the following if him and told him of my interest in analysis in he were alive today, but I think it is fair to say that math was his church. Taking this further, I Judith Packer is professor of mathematics at the Uni- think his proof of the existence of God was the versity of Colorado at Boulder. Her email address is intertwining of all branches of mathematics and [email protected].

August 2007 Notices of the AMS 837 general and ergodic theory in particular; he was board, and if one’s concentration dipped right enthusiastic, and I first began to get a sense that after lunch, when the class was held, one could to him analysis was a powerful influence on all ar- entertain oneself by reading the names “Bernoul- eas of mathematics, just as other areas influenced li”, “Euler”, “D’Alembert”, “Lagrange”, “Laplace”, the development of analysis. He immediately gave “Legendre”, etc., outlined in mirror symmetry me the galley proofs of his book Unitary Group across his jacket. Representations in Physics, Probability, and Num- Mackey was fascinated by this intertwining of ber Theory that had just been published [4] and different areas for as long as I knew him, and directed me to the chapter in question. I was whenever he gave a lecture, be it public or pri- stunned, of course: I had thought I knew a lot vate, one could see that he was totally immersed about an area, and I began to see how I just knew in the mathematics that he so loved. His private a lot about a teeny speck in a corner of one of lectures were just as well prepared and thought- the chapters of the book. From a small subset out as his public ones. Whenever he came back of shift-invariant subspaces, I stared down into from an important conference, he would give the deep ocean of locally compact group actions me a series of lectures on the main topic of the on standard Borel spaces, neither of which, the day. “K-theory for C∗-algebras! You must learn groups nor the spaces, I knew much about. Also, K-theory for C∗-algebras!” he said after returning Mackey’s use of ergodic theory in the Unitary from Europe one August, and he proceeded to Group Representations book was aimed towards give me a private series of three lectures that he an understanding of certain quasi-orbits arising had carefully crafted himself. from the theory of induced representations. Go- All the professors at Harvard were world ing back to the drawing board, I told him I needed famous of course, and all were powerful math- to know more analysis; I had studied most of ematicians, including the Benjamin Peirce Assis- first-year analysis, but knew nothing about the tant Professors. At the time I was in graduate spectral theory of normal operators. Well, he said, school, the department was particularly well why didn’t I grade homework for the first-year known in the areas of algebraic geometry and analysis course he was teaching? It would be a number theory. Because of this and because of good way to earn a bit of pocket money and learn Mackey’s “telescope rather than microscope” more analysis at the same time, he opined, and we approach, a few “youngsters” did not know of agreed that when the lectures arrived at spectral his technical prowess. I remember a colleague theory, I could leave my grading duties aside and was shocked when he was informed that George take notes. I did as he suggested, feeling a bit had been one of the first Putnam fellows. This guilty on behalf of the students enrolled in the surprise would have been avoided if he had read course, but I knew I was fortunate. some of Mackey’s deep papers on Borel spaces, The spring semester of my first year in grad- for example “Borel structure in groups and their uate school I took a course on the history of duals”, appearing in 1957 [2], or “Point real- harmonic analysis from Professor Mackey. The izations of transformation groups”, appearing point of view in this course, as it was in all of in 1962 [4], both showing a total mastery of his mathematics, was that harmonic analysis powerful techniques needed for specific results. gave a unified way to attack problems in physics, A few graduate students of my era, maybe probability, and number theory. His approach because they were in a different area and did not to mathematics was much more broad than any know much of his work, wondered why Mackey I had experienced before. Rather than focusing did not support the department by getting Na- on any specific problem and performing narrow tional Science Foundation (NSF) grants. At that problem solving (which he must have had an time, George’s opposition on principle to accept- obvious talent for, being one of the first Put- ing grant money might not have been widely nam fellows), he preferred to give a very broad known among graduate students. This was just overview so that we could view everything from five years after Watergate and the end of the Viet- a high perch looking down. “I want to use a tele- nam War, but only one or two years prior to the scope, not a microscope!” he said. He felt deeply election of Ronald Reagan. Not many people at that so many aspects of mathematics were “seem- the time were suspicious of the NSF’s control over ingly distinct, yet inseparably intertwined.” Every research, which George had predicted decades lecture in this course would start out with him earlier could arise. He told me that he found it writing down the names of five, six, sometimes impossible to write up a plan saying what he was up to twelve, famous mathematicians who were going to do in the future with any precision and contemporaries. One would think he would never to write out progress reports to explain to what be able to get through all of the mathematicians extent he had stuck to his “research plan”. He in one lecture, but by the end of the lecture he also did not see any reason why an organization would have connected them all. His enthusiasm funded by the government should need such a was such that he would often lean against the plan anyway. “I will go where the mathematics

838 Notices of the AMS Volume 54, Number 7 leads me,” he said. “Why would I follow any pre- conditions, this representation will be an assigned plan if I found something even better?” irreducible representation of G. He did not seem to get involved in the interde- This procedure provided up to unitary equiv- partmental politics of academe and completely alence all possible irreducible representations enjoyed his mathematics and his world travels, of G. One sees here how Mackey used the in- where he had an amazing amount of recognition duction process, the action of the group K on at other institutions in the U.S. and overseas. the Borel space N,ˆ and the theory of projective I want to discuss a little bit of George’s math- representations—all were unified towards the so- ematics that most influenced me. The paper lution of the problem of describing the structure “Unitary representations of group extensions, I” of G.ˆ With the Mackey machine, new technical [3], which appeared in 1958, was a beautiful com- difficulties arose in studying the action of K on bination of theory and technical prowess used Nˆ; this was one reason for his initial interest towards the goal of understanding the theory of in ergodic theory. Moreover, projective repre- unitary representations. In this particular paper a sentations arose very naturally in mathematical variety of different areas of his expertise were ap- physics, so it was not at all surprising that he parent: induced representations, the use of group was drawn to projective multipliers on locally actions on standard Borel spaces, and projective compact groups. I think this paper is characteris- representations. He formalized a method, the tic of George’s method of drawing together tools “Mackey machine”, which allowed one to deduce from seemingly disparate areas towards giving a all the equivalence classes of unitary representa- solution of a particular problem. At the end of tions of a locally compact group G if one knew this and many other papers, he was often more them a priori for a closed normal subgroup N of intrigued by any new questions that had arisen G (e.g., if N were abelian) and if G/N acted on while solving the given problem. His approach led Nˆ = {unitary equivalence classes of unitary to certain developments in the theory of operator representations of N} in a nice enough way. algebras; in particular the study of the structure Indeed, Mackey showed in the above article of certain crossed product C∗-algebras was very that if N were type I and regularly embedded in much influenced by the “Mackey machine”. G, then the set-theoretic structure of the dual Meanwhile, to jump forward in time a bit, I kept of G could be described completely in terms of attending George’s courses in graduate school, N,ˆ G/N,[ and certain projective dual spaces of even those in topics where my background was subgroups of G/N. In such a case, the Mackey less than ideal. He encouraged me to attend his machine can roughly be described as follows: course on quantum mechanics, and when I remarked that I had just had one year of college (1) Fix χ ∈ N.ˆ physics, he said that was plenty for what he would (2) Let K = G/N; then K acts on Nˆ via g · do. I still remember when he told us in an excited = −1 χ(n) χ(gng ). Let Kχ be the stabilizer manner that the mathematics behind quantum = −1 subgroup of χ, and let Gχ π (Kχ). field theory could explain “. . . why copper sulfate The group Kχ is called, following E. Wign- was blue! Boiling points, colors of chemicals, all of er, the “little group”. these can be worked mathematically!” I remember (3) Extend χ from a unitary representation being particularly struck by the fact that some of N to a possibly projective unitary chemical compound was blue precisely because representation on L on Gχ with a repre- of the mathematics rather than the physics. This sentation space Hχ. The representation χ appealed to me greatly, because it gave me hope of N uniquely determines the equivalence that I could understand a bit of physics after all class of the multiplier σ on Gχ. This is if I only could learn the mathematics. the “Mackey obstruction”. One can choose I continued to lurk in George’s peripheral vi- things so that σ is a lift of a multiplier ω sion, and both of us gradually realized that I on the quotient group Kχ. wanted to be his student. When that propitious (4) If ω is nontrivial, let M be any irreducible moment came, he went to the corner of his office ω representation of Kχ and lift it to an and picked up a huge stack of preprints from the irreducible representation of Gχ on the previous five years or so. “I was going to throw Hilbert space Hσ , also denoted by M. these away, because I have the offprints now, but The tensor product representation L ⊗ M maybe you will find something interesting here,” will then be an irreducible (nonprojective) he said. I hauled the stack of thirty or so preprints unitary representation of Gχ. in various stages of development off to my cubi- (5) There may be many such choices, depend- cle, wondering if this would be like looking for a ing on the σ -representation theory of Gχ. needle in a haystack. But I did look through all of (6) Form the induced representation them; it turned out there were many needles and G ⊗ IndGχ (L M). Again, under appropriate hardly any strands of hay in the pile. I still have

August 2007 Notices of the AMS 839 most of these preprints in my files, and most were partially for the enjoyment of provoking. Partly written by a variety of luminaries (E. Effros, C. C. for this reason, a few fellow graduate students, Moore, A. Ramsay, M. Rieffel, M. Takesaki, C. Se- not knowing George well and only seeing him at ries, R. Zimmer, to name just a few) whom I later teatime, thought that because of his age (at that had the good fortune to be able to meet in my time in his mid-sixties) and his outward manner, postdoc year at UC Berkeley and during trips to he must have been “sexist” as an advisor. I told UCLA, Penn, and Colorado through the years. As them that nothing was further from the truth, time went on and I became more mathematically that all he cared about was his students’ interest mature, I realized that George had given me a and abilities in mathematics, and what he most gold mine of information. He turned out to be an enjoyed doing was talking about mathematics to excellent advisor, one who gave me wide latitude anyone, male or female. Benbow and Stanley and to think about and work on the thesis problem their ilk were transitory, but mathematics was that was most interesting to me, in my case a forever, mathematics was pure, mathematics was study of the relationship between the structure sacrosanct. I think that one of his students that of von Neumann algebras and some subalgebras he mentioned with the most pride while I was constructed using ergodic actions and quotient a student was Caroline Series (now a professor actions. He gave me many great leads and math- at Warwick and also contributing to this article), ematical ideas on my thesis, and on other of my and when I had the opportunity to meet her, she papers because of his all-encompassing knowl- seemed to share some similar sentiments about edge of the literature and understanding of the George. When it came to the fundamentals of big picture. Whenever I came in with a question advising and mentoring graduate students, by about something, even if I was not able to formu- helping and encouraging them, all that mattered late exactly what I needed to know, he seemed to to him was the mathematics they were doing; in know what I needed. I remember querying him my opinion, in the treatment of women mathe- in a confused fashion about masa’s (maximal maticians in his own way he was far ahead of abelian self-adjoint subalgebras) of von Neumann many people many years his junior. He was very algebras. “Well, do you know about the paper of comfortable with women’s abilities, and with Singer? You must read the paper of Singer!” and anyone’s abilities for that matter, whenever they he zipped out of his office on the third floor to talked to him about mathematics. This may have the conveniently located math library just outside been the case because his wife, Alice, was so ac- of it, and found the paper by I. M. Singer [6] in complished and because together they had such a question in a jiffy. As he had indicated, it was brilliant daughter, Ann, whom he introduced with extremely insightful and useful, and invaluable in great pride to me when she was a junior in high my thesis work. school at the department winter holiday party. Since many people have asked me through the I now arrive at his personal kindness. Whenever years, I feel here I should address George’s views on women and their abilities in mathematics. At I was discouraged about my mathematical abili- about the same time that I began working with ty, which was often in the early years following George, an article by Camilla Benbow and Julian my Ph.D., he would always sound a positive note, Stanley had just appeared [1], and the topic of telling me of some stellar mathematician who had whether or not most females aged twelve and been similarly discouraged and gone on to great above were indisposed by virtue of their sex to do achievements. I know of others who tell similar deep mathematics was splashed across the front stories of his kindness and deep sense of loyalty page of all the newspapers. The interpretation to all of his colleagues and students, regardless of of the contents of this article became a thorn their fame or fortune. in the side of women mathematicians in general It was fun to catch up with George throughout and women mathematics graduate students in the years, first in 1983, right after my postdoc particular. Of course this was talked about at the year in Berkeley, and finally in the summer of daily teas in the Harvard math department, and, 1999, when my husband, my two young boys, as Mackey was a constant presence at teatime, and I visited George and Alice in their Coolidge he would discuss these results just as he would Hill Road house. They both showed us great any other topic of the day. He would ask the hospitality, and George told me of his latest en- graduate students what they felt and would muse thusiastic mathematical ideas about finite simple on whether or not this article provided some sort groups and also about his great pride in his young of “proof” of anything and, in fact, could one grandson. obtain a “proof” of this sort rather than merely I believe that George was one of the great make observations, which he proceeded to make. mathematicians and great mathematical char- I was very annoyed of course; it took me some acters of the past century. I can see him in my time to realize that he talked about this partially mind’s eye now, with his blue seersucker jacket, for the enjoyment of provoking discussion and often with chalk dust on the back of it, with a

840 Notices of the AMS Volume 54, Number 7 smile on his face as he is describing the latest ad- physics to mathematics, and from then on I think vance in mathematical research that either he has I took every course George gave, and our meals uncovered or heard about in the latest conference together became even more frequent. he has attended. It is hard to think of this world In my senior year, George said he felt that without George in it. I will miss him greatly. I should have some diversity in my mathematical educational experience and advised me to References go elsewhere for my graduate training. But this [1] C. P. Benbow and J. C. Stanley, Sex differences in was one piece of advice from him that I ignored, mathematical ability: Fact or artifact?, Science 210 and I was very glad for his continued vital help (1980), 1262–1264. and encouragement as I worked toward my Ph.D. [2] G. W. Mackey, Borel structure in groups and their degree at Harvard. While Andy Gleason became duals, Trans. Amer. Math. Soc. 85 (1957), 134–165. my thesis advisor (on Mackey’s advice), George [3] , Unitary representations of group exten- was still an important advisor and mentor dur- sions, I, Acta Math. 99 (1958), 265–311. ing my graduate years, and in the Mathematics [4] , Point realizations of transformation groups, Illinois J. Math. 6 (1962), 327–335. Genealogy Project I added him as my second ad- [5] , Unitary Group Representations in visor. And George also played an important role Physics, Probability, and Number Theory, Ben- in getting me an instructorship at the University jamin/Cummings Publishing Co., Boston, MA, of Chicago, then the preeminent place for a first 1978. academic job. He had many friends there, and he [6] I. M. Singer, Automorphisms of finite factors, put in a personal good word for me with them. Amer. J. Math. 77 (1955), 117–133. Experiences such as mine were repeated over and over with many other young mathematicians just starting their careers. For while George was Richard Palais totally devoted to his mathematical research, he was never so busy that he could not spare some George Mackey had a pivotal influence on my life: time to help a student or younger colleague in my contacts with him, early and late, determined their studies and their research. who I was, what I would become, and how my life George and I renewed our friendship when I and career would play out. He was in many ways a returned to the Boston area in 1960 to take a model for me, and throughout my career as math- faculty position at Brandeis, and over the next ematician and teacher I have frequently realized forty years, unless one of us was away on sab- that in some important decision I made or in some batical, we made a practice of meeting frequently way that I behaved I was attempting to emulate for lunch in Harvard Square and discussing our him. If I were an isolated case, this would hardly respective research. This gave me a good and per- be worth noting here, but there is considerable ev- haps unique perspective of George’s long-term idence and testimony that George influenced very research program and goals as he himself saw many others with whom he had contact in similar them. So, although I will leave to others in accom- ways. panying articles any detailed review or appraisal I first got to know George in 1949. I was then of his research, I feel it may be of some interest in my sophomore year at Harvard and took his if I comment on his research from this unusual famous Math 212 course. It started from the perspective. most elementary and fundamental part of math- If you asked Mackey what were the most im- ematics, axiomatic set theory, proceeded through portant ideas and concepts that he contributed, I the development of the various number systems, think he would have cited first the circle of ideas and ended up with some highly advanced and around his generalizations to locally compact esoteric topics, such as the Peter-Weyl Theorem. groups of the imprimitivity theorem, induced It was exciting and even breathtaking, but also representations, and Frobenius reciprocity for very hard work, and I spent many hours following finite groups, and secondly what he called Virtual each lecture trying to digest it all. But what made Groups. Indeed, his 1949 article “Imprimitivity it a truly life-altering experience was that George for representations of locally compact groups I” was also a tutor in my dorm, Kirkland House, in the Proceedings of the National Academy of and lived there himself. He encouraged me to Sciences was a ground-breaking and very highly take several meals with him each week, where we cited paper that gave rise to a whole industry of discussed my questions about the course but also generalizations and applications, both in pure about mathematics and life in general. By the end mathematics and in physics. George himself of that year I decided to switch my major from used it as a tool to analyze the unitary repre- Richard Palais is professor emeritus at Brandeis Uni- sentations of semidirect products, and it was versity and adjunct professor of mathematics at the an essential part of the techniques that, in the University of California at Irvine. His email address is hands of Armand Borel and Harish-Chandra, led [email protected]. to the beautiful structure theory for the unitary

August 2007 Notices of the AMS 841 representations of semisimple Lie groups. But Arlan Ramsay Virtual Groups, introduced in his paper “Ergodic theory and virtual groups” (Math. Ann. 186, 1966, Remembering George Mackey 187–207) were another story. Even the name did not catch on, and they are now usually called Like so many others, I have benefited often from groupoids, although that name is used for a the clarity of understanding and of exposition of great many other closely related concepts as well. George W. Mackey. If this had been only via his Mackey always felt that people never fully under- publications, it would have been quite stimulating stood or appreciated his Virtual Group concept and valuable, but it has been a truly wonderful op- and believed that eventually they would be found portunity to learn from him in courses and series to be an important unifying principle. In the later of lectures and even on a one-to-one basis. years of his life he tried to work out his ideas in Moreover, after a time, our relationship became this direction further. one of comfortable friends. My wife and I have Another aspect of his research about which both enjoyed knowing George and his delightful Mackey was justifiably proud was his contri- wife, Alice. The pleasures have been substantial butions to developing rigorous mathematical and greatly enjoyed. foundations for quantum mechanics. His 1963 As I revise this, I am visiting at the Institut Hen- book Mathematical Foundations of Quantum Me- ri Poincaré, where there have been numerous talks chanics was highly influential and is still often about uses of groupoids in physics and geometry, quoted. (The title is a direct translation of von particularly noncommutative geometry. If George Neumann’s famous and ground-breaking Math- were here, he would make wonderful reports on ematische Grundlagen der Quantenmechanik, the activities. Even with his example to learn from, which George told me was intentional.) His work I can achieve only a poor imitation. Still, I am hap- in this area stops with the quantum theory of py to have the example as a standard. particle mechanics, and several times I suggested My first encounter with George was in the un- that he go further and develop a rigorous mathe- dergraduate course he taught in 1958–59 on pro- matical foundation for quantum field theory. But jective geometry. Already I found his attitude and he seemed rather dismissive of quantum field style appealing. theory and, as far as I could tell, was dubious that Then I was in the class in the spring of 1960 it was a valid physical theory. for which he wrote the notes that became the George was a great believer in what Wigner book Mathematical Foundations of Quantum Me- called “the unreasonable effectiveness of Math- chanics [M1963]. The book by John von Neumann ematics in the natural sciences.” I want to close of the same title had aroused my curiosity about with a quote from a lecture that Mackey gave the subject, so this course was clearly a gold- that I feel epitomizes his feeling of wonder at en opportunity, but the reality far exceeded my the beauty and coherence of mathematics, a feel- expectations. ing that motivated his whole approach to his With his customary thoroughness, Mackey research. began the course with a discussion of classical While it is natural to suppose mechanics in order to be able to explain the that one cannot do anything very quantizing of classical systems and the need useful in tool making and tool for quantum mechanics. He explained about the improvement, without keeping problem of blackbody radiation and Planck’s idea a close eye on what the tool is for solving the problem, along with other pre- to be used for, this supposition cursors of quantum mechanics. We heard about turns out to be largely wrong. the indistinguishability of electrons and the idea Mathematics has sort of inevitable that electrons are all associated with a greater structure which unfolds as one totality, like waves in a string are configurations studies it perceptively. It is as of the string. There were many such insights to go though it were already there and along with the discussion of axioms for systems one had only to uncover it. Pure of states and observables that exhibit appropriate mathematicians are people who behavior to be models for quantum mechan- have a sensitivity to this structure ics. This course answered many questions, and and such a love for the beauties then the answers raised further questions. It was it presents that they will devote just what was needed and gave me a start on a themselves voluntarily and with long-term interest in quantum physics. enthusiasm to uncovering more George had invested a great deal in support of and more of it, whenever the his aim to understand quantum mechanics. He opportunity presents itself. spoke of reading parts of the book by Hermann

842 Notices of the AMS Volume 54, Number 7 Weyl, The Theory of Groups and Quantum Me- to the list of distinguished speakers. He gave chanics, and then working at length to explain the outstanding lectures on the uses of symmetry, material in his own framework. Having done that, particularly on the application of finite groups to he was happy to pass along his understanding in the spectra of atoms. the course and the notes. He was an example of Then at MSRI in 1983–1984, he gave a long se- the best practices in communicating mathemat- ries of lectures about number theory. As was his ics, always ready to be a student or a teacher, as habit, the historical background was an important appropriate. feature, and symmetry was the star of the show. A The course George taught in the academic year dinner at the Mackeys’ was one of the memorable 1960–1961 was on unitary representations of lo- social events for us. cally compact groups. He also gave an informal Almost twenty years later, in November of seminar on the historical roots of harmonic anal- 2002, George had an interest in symmetry that ysis. His constant interest in historical relation- was as intense as ever. He was also interested ships was well exhibited in the course and those in discussing a variety of other topics: human lectures. nature, recently published books, etc. That was He connected representation theory to quan- our last conversation, and I wish there could be tum mechanics by way of the symmetry one many more. Regarding mathematics and physics, might expect for a single particle in Euclidean what interested him most at that time was the 3-space. The “total position observable” of a par- potential uses of symmetry. ticle in R3 should be a system of imprimitivity for Regarding George’s papers, of special interest a representation of the Euclidean group E as ex- 3 to me was [M1958]. In it George investigated the plained in [M1968] and in Chapter 18 of [M1978]. unitary representations of G, where G is a (sec- The basic reason is that a Euclidean motion can ond countable and) locally compact group and N be regarded as a change of coordinates, and mea- is a type I normal subgroup. The idea is to gain surements in one system should correspond to some information from the presence of N and the measurements in another in a systematic way. way G acts on N and hence the unitary dual of If Q is the question whether the particle is in a E N, Nˆ (unitary equivalence classes of irreducible Borel set E and g ∈ E3, the relationship desired is that representations of N). Composing with inner automorphisms of G restricted to N gives a natural Q = U(g)−1Q U(g). Eg E action of G on Nˆ. He proved that if L is a factor Moreover, if the particle is to be treated in representation of G, then the canonical decompo- the framework of special relativity, then U must sition of L|N over Nˆ uses a measure class [µ] that extend to the appropriate group of space-time is quasiinvariant and ergodic for that natural ac- symmetries, the Poincaré group. By using Mack- tion. Call [µ] the measure class associated with ey’s analysis of representations of group ex- L. In many cases, N is what he called regularly tensions [M1958], such representations of the imbedded ; i.e., every measure class on Nˆ that is Poincaré group can be classified, and the mass ergodic and quasiinvariant for the action of G is and spin of the particle appear from the analysis. concentrated on an orbit. This approach to particles as objects that can The proofs used in [M1958] end by using tran- be localized was also used by A. S. Wightman in sitivity, i.e., by working on a coset space and tak- [W1962]. The connection to physics was only one of a ing advantage of the existence of a stabilizer that number of reasons that Mackey was so interested carries the information that is needed. However, in understanding and exploiting symmetry. Num- his proof of the Imprimitivity Theorem in particu- ber theory, probability, and ergodic theory also lar begins in a way that works for general quasiin- came under that umbrella [M1978]. variant measures, and his method for getting past Over the years I had several other opportuni- the sets of measure 0 can be adapted to the gen- ties to learn from George’s unique perspective in eral case. person. The first instance was a visit to Cambridge Suppose that [µ] is a measure class concentrat- in the mid 1960s, and the next was at a confer- ed on an orbit in Nˆ and that H is the subgroup of ence in 1972 at TCU, organized by Robert Doran. G/N stabilizing a point on that orbit. Then all the There was a great deal of excitement about the re- representations of G whose associated class is [µ] lationship between virtual groups and orbit equiv- can be expressed in terms of the (possibly multi- alence in ergodic theory. George was delighted to plier) representations of H/N (see the contribu- learn of earlier results of Henry Dye about orbit tions above by C. C. Moore and J. Packer). More- equivalence for countable abelian groups. over, every representation produced by the con- During the following academic year, George struction is irreducible. The Imprimitivity Theo- gave the De Long Lectures at the University of rem plays a key role in the analysis. If N is not Colorado in Boulder, adding his own distinction regularly imbedded, the required information is

August 2007 Notices of the AMS 843 not contained in individual orbits, but more gen- [M1968] , Induced Representations of Groups and eral quasiorbits. Quantum Mechanics, W. A. Benjamin, Inc., New In the more general case, George recognized York-Amsterdam; Editore Boringhieri, Turin, 1968, that the action of G on Nˆ can be used to give Nˆ ×G viii+167 pp. [M1978] , Unitary Group Representations in a groupoid structure and that the first step in his Physics, Probability, and Number Theory, proof of the Imprimitivity Theorem in [M1958] Mathematics Lecture Note Series, 55, Ben- constructs a unitary valued function that satisfies jamin/Cummings Publishing Co., Inc., Reading, the equation defining a groupoid homomorphism MA, 1978, xiv+402 pp. almost everywhere on Nˆ × G. This fact motivated [R1976] Arlan Ramsay, Nontransitive quasiorbits in his introduction of ergodic groupoids and their Mackey’s analysis of group extensions, Acta similarity equivalence classes, the latter being Mathematica 137 (1976), 17–48. A. S. Wightman what he called virtual groups. His plan was to use [W1962] , On the localizability of quantum mechanical systems, Rev. Modern Physics 34 virtual subgroups of G to replace the stabilizers (1962), 845–872. that appear in the regularly imbedded case. In that case Nˆ × G is equivalent to the stabilizer of any point in the orbit carrying [µ], and it follows Caroline Series that the representation behavior for Nˆ × G is exactly the same as it is for the stabilizer. Of course, changing the choice of point in the orbit George Mackey changes the stabilizer to a conjugate subgroup, In my memory, George Mackey scarcely changed so the action itself is equivalent to a conjugacy from the time I first approached him as a prospec- class of subgroups. Since a virtual group is an tive graduate student in the mid-1970s until the equivalence class of groupoids, it is essentially last time I saw him a couple of years before his inevitable that one tends to work with groupoids death. He was a scholar in the truest sense, his themselves. There are also other kinds of equiv- entire life dedicated to mathematics. He lived a alence besides similarity, and groupoids persist life of extraordinary self-discipline and regular- in all the contexts. The fundamental idea remains ity, timing his walk to his office like clockwork the same. and managing, how one cannot imagine, to avoid I want to say something about the existence teaching in the mornings, this prime time being of [R1976]. I inquired about whether George was devoted to research. I was once given a privileged planning to write the continuation of [M1958], view of his study on the top floor of their ele- and George generously encouraged me to carry gant Cambridge house. There, surrounded on all out that project myself. sides by books and journals shelved from floor Those results were combined with other work to ceiling, he had created a private library in to show that it is possible to transfer information which he immersed himself in a quiet haven of from one nonregularly imbedded group extension mathematical and intellectual scholarship. I was to another in [BMR] and to investigate nonmono- lucky to arrive in Harvard at the time when he mial multiplier representations of abelian groups was working on what subsequently became his in [BCMR]. Of course finding all representations famous Oxford and Chicago lecture notes [7, 8]. for a given virtual group is virtually impossible, These masterly surveys convey the sweep of huge but useful information can be obtained. George’s parts of mathematics from group representations notion of virtual group is now thriving in a num- up. I do not know any other writer with quite his ber of areas, and representation theory remains gift of sifting out the essentials and exposing the part of the program. bare bones of a subject. There is no doubt that his unique ability to cut through the technicali- References ties and draw diverse strands together into one [BMR] L. W. Baggett, W. Mitchell, and A. Ramsay, grand story has been a hugely wide and enduring Representations of the discrete Heisenberg group influence. and cocycles of an irrational rotation, Michigan Mackey believed strongly in letting students Math. J. 31 (1984), 263–273. find their own thesis problem. Set loose reading [BCMR] L. W. Baggett, A. L. Carey, W. Moran, and his notes, I reported to him every couple of weeks, A. Ramsay , Non-monomial multipler representa- and he was always ready to point me down some tions of Abelian groups, J. Funct. Anal. 97 (1991), new yet relevant avenue. He never helped with 361–372. [M1958] George W. Mackey, Unitary representations of technical problems, always saying, “Think about group extensions I, Acta Mathematica 99 (1958), it and come back next week if you are still stuck.” 265–311. [M1963] , The Mathematical Foundations of Caroline Series is professor of mathematics at the Quantum Mechanics, W. A. Benjamin, Inc., New University of Warwick, UK. Her email address is York-Amsterdam, 1963. [email protected].

844 Notices of the AMS Volume 54, Number 7 Sometimes I envied the other students, whom I, rather s · g, since Mackey insisted on doing all his somewhat naïvely, assumed were being told ex- group actions from the right). Thus (s, g) could actly what to do, but in retrospect this was a most be composed with (s′, g′) if and only if s′ = s · g valuable training. Stemming from his interest and then (s, g) ◦ (s′, g′) = (s, gg′). in ergodic theory, I was given a partially guided The next step is perhaps the most interesting: tour of a wide swathe of dynamical systems.This a homomorphism from S ×G to a group H should loose but broad direction stood me in good stead be a cocycle: that is, a map a : S × G → H such later; I often think of it when under pressure to that a(s, g)a(sg, g′) = a(a, gg′). This led Mackey hand out precisely doable thesis problems when to his construction of the “range of the homomor- students have barely started their studies. phism”. Observe that, in deference to Mackey’s I finally settled for working on Mackey’s won- order, the direct product G × H acts on S × H derful invention, virtual groups. The idea, touched by (s, h) · (g,h′) = (sg,h′−1ha(s, g)). The “range” on in C. C. Moore’s article, is laid out in most de- is essentially the action of H on the space of G tail in Mackey’s paper [5]. His explanation to me orbits: if this space is not a standard Borel space, was characteristically simple. He was interested as, for example, if the G action is properly ergod- in group actions on measure spaces, because a ic, one replaces it by the largest standard Borel measure-preserving action of a group induces a quotient. Mackey saw this as the generalisation natural unitary representation on the associated of the dynamical systems construction of a “flow L2 space. As with any class of mathematical ob- built under a function”. (A cocycle for a Z-action jects, he said, if you want to understand group can be defined additively given a single function actions, you should split them up into the sim- from S to H.) This circle of ideas was seminal for plest possible pieces. The simplest kind of group much future work, in particular that of Mackey’s action is a transitive one, that is, one with a single former student Robert Zimmer. A special case is orbit. Being a generalist, Mackey wanted to work the Radon Nikodym derivative of a measure class in the category of Borel actions, so in particular preserving group action, which can be viewed as the groups he cared about always had a standard a groupoid homomorphism to R, of which more Borel structure, that is, were either countable or shortly. Borel isomorphic to the unit interval. As he point- A special case arises when all the stabilisers ed out, an action being Borel has unexpectedly of points are trivial. Mackey’s natural relation of strong consequences; in particular the stabilisers similarity between groupoids leads to the clas- of points are closed. A transitive action is clearly sification of free ergodic actions up to “orbit determined by the stabiliser of a single point equivalence”. Two measure-class-preserving ac- or, more precisely, since the choice of point is tions of groups G, G′ on spaces S, S′ are called arbitrary, by a conjugacy class of such. Thus a orbit equivalent if there is a Borel measure-class- transitive action of a Borel group on a standard preserving map φ : S → S′ with the property Borel space is equivalent to the specification of a that two points in the same G-orbit in S are conjugacy class of closed subgroups. mapped to points in the same G′-orbit in S′. What is the next simplest type of action? Since (This is much weaker than the usual notion of we are in a category of measure spaces, an “in- conjugacy, in which one insists that G = G′ and decomposable” action means that the underlying that φ(sg) = φ(s)g.) Just how much weaker is space should not split into nontrivial and measur- expressed in a remarkable theorem discovered able “subactions”. Assuming the group preserves by Mackey’s student Peter Forrest: any two finite a measure or at least a measure class, this is measure-preserving actions of Z are orbit equiv- precisely what is meant by the action being er- alent [3]. Subsequently, Mackey learnt that the godic: there are no “nontrivial” group invariant theorem had previously been proved by H. Dye measurable subsets, where “nontrivial” means [2] and became a great publicist. He took plea- neither a null set nor the complement of a null sure in telling me it had also been proved by the set. Mackey’s idea was that, since a transitive Russian woman mathematician R. M. Belinskaya action is determined by a closed subgroup, then [1]. More generally, any equivalence relation or- wouldn’t it be nice if an ergodic action were bit equivalent to a Z-action is called hyperfinite. similarly determined by a new kind of “subob- Dye’s theorem extends to show that actions of a ject” of the group, which he named in advance much wider class, including all abelian groups, a “virtual group”. There is a touch of genius in are hyperfinite, culminating in Zimmer’s result his passage from this apparently simplistic idea [9] that an action is hyperfinite if and only if the to a formal mathematical structure yielding deep equivalence relation is amenable in a suitable insights. The key point is that the sought-after sense. virtual group should be the groupoid S × G, in The classification of non-measure-preserving which the base space is the underlying space of Z-actions up to orbit equivalence is even more the action S and the arrows are pairs (s, g) where remarkable. Regarding the Radon Nikodym deriv- (s, g) has initial point s and final point g · s (or ative as a cocycle to R as above, Mackey’s “range”

August 2007 Notices of the AMS 845 is known to dynamicists as the Poincaré flow. At the time Mackey did not perhaps appreciate Eulogy by Andy Gleason just how far-reaching a construction this was. If It was nearly sixty years ago that I first met Z the original -action is measure-preserving, it is George. The circumstances were that I had called Type I, II0, II∞, depending on whether the just arrived here in Cambridge myself, and I R range -action is transitive or preserves a finite started to listen to his course. He was then or infinite measure respectively. If the original an assistant professor, and he gave a course action only preserves a measure class, it is called on locally compact groups, something he did Type III1, IIIλ, and III0, depending on whether many, many times thereafter—a course he Z R+ the range groupoid is {id},λ for some λ ∈ , or gave often. properly ergodic. In a beautiful and remarkable And I went to his course and listened, piece of mathematics, pushed to its conclusion and pretty soon we started having after-class by W. Krieger [4], it turns out that the range conversations. And then I began a policy of Z completely classifies the original -action up to frequently going around to his rooms and orbit equivalence. The same construction gives talking mathematics. Just about general parts rise to a rich fund of examples of von Neumann of mathematics. Not about his course partic- algebras, a fact widely exploited by A. Connes. My ularly, maybe a little of this, maybe a little of training under Mackey was an ideal foundation that, but whatever it was. from which to appreciate all this work, which was And I, in any event, learned a great deal developing rapidly in the late 1970s just about from these conversations. And yet, when I the time I finished my thesis. look back, I can’t really focus on a specific thought that comes out of those conversations. It was a very general thing; it’s because when we talked, we didn’t talk about mathematics at the level where you see it in publications, with the revolving theme of first definition, then a theorem, then a proof. We didn’t do that; we talked in very specula- tive terms. And I just found that I learned a great deal from this kind of conversation, and I’m really very deeply indebted to George for having told me these things—many, many things. We went on having those conversations for the next two years, and then he went away to France for a year, and when he came back we started up again. Then I was away for two years, and we had to start again. Finally, we used to talk after we were mar- Photo courtesy of Richard Palais. ried and I and he both lived out toward the Andrew Gleason (left) with Mackey at an 80th west end of Cambridge. We used to walk birthday party for Alice Mackey. along Brattle Street, continuing the conversations of then, by this time, twenty years To return to Mackey as a person. Everyone who before. knew him will remember his uncompromising And they were the same kinds of things. and sometimes uncomfortably forthright intel- They were speculations about the nature of lectual honesty. He took pleasure in following mathematics. They weren’t theorems about through a line of thought to its conclusion: politi- mathematics; they were just speculations. cal correctness was not for him. He wrote several And some of them worked out ultimate- articles about the invidious effects of federal ly, some of them did not, and that’s the research funding [6]. He may have lost the battle, way mathematics is. And I very much ap- but what he said was quite true. preciate the fact that I learned a great deal I must say something about the widely held from George and I went on from there, and view that Mackey was against women mathemati- eventually was able to become his colleague. cians. All that I can say is that I never experienced the slightest prejudice from him and am proud to Thank you. be what he referred to as “his first mathematical daughter.” His straightforwardness was perhaps Andrew Gleason easy to misinterpret. For example, he might say something like “There have been historically

846 Notices of the AMS Volume 54, Number 7 almost no women mathematicians of stature. [5] G. Mackey, Ergodic theory and virtual groups, Therefore, on a statistical basis it is unlikely that Math. Ann. 166 (1966), 187–207. a particular woman will be one.” Of course such [6] , Effort reporting and cost sharing of federal a view ignores all the complicated historical and research, Science 153 (Sept. 1966), 1057. [7] , The Theory of Unitary Group Representa- cultural factors which explain why this might be tions, Univ. of Chicago Press, 1976. so; nevertheless, the fact was hard to dispute. But [8] , Unitary Group Representations in Physics, what remains in my memory is that Mackey was Probability and Number Theory, Benjamin, 1978. always open-minded and unprejudiced, willing to [9] R. J. Zimmer, Hyperfinite factors and amenable take on-board new insights or experiences and actions, Invent. Math. 41 (1977), 23–31. accept new people on their merits, exactly as they came along. Mackey stood for the highest standards. He V. S. Varadarajan lived his life by precise rules, but he enjoyed it to the fullest. Within his mathematics he found a I first met George Mackey in the summer of 1961 fulfilment to which few can aspire. The spartan when he visited the University of Washington at simplicity and mild disorder of his office con- Seattle, WA, as the Walker-Ames professor to give trasted sharply with the comfortable elegance a few lectures on unitary representations and on created by Alice in their Cambridge home. Her quantum mechanics. The lectures were a great wonderful old-world dinner parties were memo- revelation to me, as they revealed a great mas- rable occasions at which it was a privilege to be a ter at work, touching a huge number of themes guest. Mackey was sometimes disarmingly open and emphasizing the conceptual unity of diverse about his family life, but through it all shone topics. I had the opportunity to meet at leisure human warmth and love: their strictly scheduled with him and talk about things, and his advice but vastly important time reading aloud in the was invaluable to me. I was just getting start- evenings, his pride in their daughter, Ann. ed in representation theory, and his suggestion that I should start trying to understand Harish- Mackey had the habit of writing lengthy letters Chandra’s vast theory (based on a “terrifying about his latest discoveries. Long after retirement, technique of Lie algebras” as he put it) was one indeed right up to a couple of years before his that gave my research career the direction and death, he continued working on various projects boost it needed. which between them seemed to involve nothing In representation theory Mackey’s goal was less than unravelling the entire mathematical his- to erect a theory of unitary representations in tory of the twentieth century. Subjects expanded the category of all second countable locally com- to include statistical mechanics, number theory, pact groups. This was decades ahead of his time, complex analysis, probability, and more. He ex- considering that all the emphasis on doing repre- plained that group representations encompassed sentation theory and harmonic analysis on p-adic more or less everything, given that starting from and adelic groups would be ten to fifteen years quantum theory one obviously had to include in the future. In the aftermath of the fusion of chemistry and thus also biology. One might argue number theory and representation theory, the that things are a little more complicated; indeed I original goal of Mackey seems to have fallen by am sure with a twinkle in his eye he would agree. the wayside; I feel this is unfortunate and that What is certain is that his ability to strip things this is still a fertile program for young people down to their essential mathematical structure to get into. It may also reveal hidden features of put a hugely influential stamp on generations the p-adic theories that have escaped detection of mathematicians and physicists. He was a man because of overspecialization. whose unique qualities, insights, and enthusiasms I should mention his work on induced repre- touched us all. sentations of finite groups. Even though this is a very classical subject, he brought fresh view- References points that were extraordinarily stimulating for [1] R. M. Belinskaya, Partitions of Lebesgue space in future research. A basic question in the theo- trajectories defined by ergodic auto-morphisms, ry is to compute the dimension of the space Funktsional. Analiz i Ego Prilozen 2, No. 3 (1968), of intertwining operators between two induced 4–16 (Russian). representations of a finite group G, induction [2] H. A. Dye, On a group of measure-preserving transformations, I, II, Amer. J. Math. 81 (1959), being from two possibly different subgroups 119–159. Hi (i = 1, 2). Mackey found a formula for this n n [3] P. Forrest, Virtual subgroups of R and Z , Adv. as a sum over the double coset space H1\G/H2 Math. 3 (1974), 187–207. [4] W. Kreiger, On ergodic flows and the iso- V. S. Varadarajan is professor of mathematics at the Uni- morphism of factors, Math. Ann. 223 (1976), versity of California, Los Angeles. His email address is 19–70. [email protected].

August 2007 Notices of the AMS 847 of local (so to speak) intertwining numbers. The they represent simultaneously measurable ob- method on the surface does not seem to apply servables, and so they are random variables on when G is infinite, for instance, when G is a Lie the same probability space, thus presenting no group. However, François Bruhat, then a student obstacle to the assumption of the additivity of at Paris and interacting with Henri Cartan and the expectation values. But if A and B do not Laurent Schwartz, realized that in order to make commute, there is no single probability space Mackey’s method work one has to represent (fol- on which they both can be regarded as random lowing a famous theorem of Laurent Schwartz) variables, and so in this case the assumption of the intertwining operators by suitable distribu- additivity is less convincing. Mackey saw that tions on G that have a given behavior under the to have the most forceful answer to the hidden action of H1 × H2 (from the left and right) and variables question, the additivity can be assumed obtained a formula remarkably similar to Mack- only for commuting observables. If E is assumed ey’s, at least when the double coset space was to be additive only for commuting observables, its restriction to the lattice L(H ) of projections finite (as it is when G is semisimple and H1 = H2 is minimal parabolic). The Mackey-Bruhat theory in H would be a measure, i.e., additive over has had a profound influence on the subject, as orthogonal projections. Conversely, any such can be seen from Bruhat’s own work on induced measure would define an expectation value that representations of p-adic groups and the later has all the properties that von Neumann demand- work of Harish-Chandra on the representations ed, except that additivity will be valid only for induced from an arbitrary parabolic and, later commuting observables. Andrew Gleason then still, his work on the Whittaker representations of showed, after Mackey brought his attention to a semisimple Lie group. the question of determining all the measures L H The lectures he gave on quantum mechanics at on the orthocomplemented lattice ( ), that Seattle were an eye-opener for me. He was the first the only countably additive measures are of 7 -→ = 1/2 1/2 person who fully understood the points of view of the form P Tr(UP) Tr(U PU ) (when dim(H ) ≥ 3); from this point on, the argument von Neumann and Hermann Weyl, made them his is the same as von Neumann’s. One can use the own, and then took them to even greater heights. Gleason result to show also that there are no two- His group-theoretic analysis of the fundamental valued finitely additive measures on the lattice aspects of quantum kinematics and dynamics L(H ) if 3 ≤ dim(H ) ≤ ∞, thus showing that was very beautiful, and he adumbrated this in a there are no dispersion-free states. This analysis number of expositions. But it was his analysis of of hidden variables à la Mackey is a significant the foundations that was very original. The basic sharpening of von Neumann’s analysis and re- question, one that von Neumann first discussed veals the depth of Mackey’s understanding of in his book Grundlagen der Quantenmechanik, the mathematical and phenomenological issues is the following: Is it possible to derive the sta- connected to this question. tistical results of quantum theory by averaging a In his view of quantum kinematics, Mackey more precise theory involving several hidden vari- formulated the covariance of a quantum system ables? In his analysis von Neumann introduced with a manifold M as its configuration space as the model-independent Expectation functional E the giving of a pair (U, P) where U is a unitary O on the space of all bounded observables and representation of a group G acting on M and O showed that if = BR(H ), the space of bounded P is a projection-valued measure. For any Borel self-adjoint operators on the Hilbert space H (the set E ⊂ M, P(E) is the observable that is 1 or model for quantum theory), then E is necessarily 0 according to whether the system falls in E or 1/2 1/2 of the form E(A) = Tr(UA) = Tr(U AU ) for not, and G is the given symmetry group or at all A ∈ BR(H ), for some positive operator U least a central extension of it (the latter is nec- of trace 1. The states, which are identified with essary, as the symmetries act as automorphisms the Expectation functionals, form a convex set, of the lattice of projections and so are given by and its extreme points are, by the above result of unitary operators defined only up to a phase von Neumann, the one-dimensional projections factor). Covariance is then given by the relations − U = Pφ, the φ being unit vectors in the quantum U(g)P(E)U(g) 1 = P(g[E]). This is exactly what Hilbert space H . If there were hidden variables, Mackey had called a system of imprimitivity for the states defined by the Pφ would be convex G in his pioneering work on the extension of the combinations of the (idealized states) defined by theory of Frobenius (of induced representations) giving specific values to the hidden variables, a to the full category of all separable locally com- contradiction. pact groups, and he was thus able to subsume The basic assumption that von Neumann made the fundamental aspects of quantum kinematics about the functionals E was their unrestricted under his own work. The story, as told to me by additivity, namely, that E(A + B) = E(A) + E(B) him in Seattle, of how he came upon this formu- for any two A, B ∈ BR(H ). If A and B commute, lation of quantum kinematics is quite interesting.

848 Notices of the AMS Volume 54, Number 7 Irving Segal, who had gone to attend a confer- couldn’t begin to live up to the material and ence, sent him (Mackey) a postcard saying that social expectations set for him by his parents, Wightman had in his lectures at the conference he felt extraordinarily fortunate to have found used Mackey’s work to discuss kinematic covari- his way to a world in which he could do what ance in quantum mechanics, and Mackey then he loved best in such a rarified atmosphere. The deduced his entire formulation with no further world opened up for him, literally and figura- help other than this cryptic postcard! Of course tively, when he discovered mathematics, and he there is more to covariance than a system of never looked back. He devoted all the time he imprimitivity—indeed, even in ordinary quantum could to his work, spending most of his days in mechanics, this formulation does not prevent his third-floor study, emerging only for meals. If position observables from being defined for the I came upstairs to look for him, I could count on relativistic photon, as Laurent Schwartz discov- seeing him slumped in his chair with clipboard ered, although we know that there is no frame in hand, lost in thought and often chewing on where the photon is at rest, and so position op- one of the buttons on his shirt. That clipboard erators cannot be defined for it. Much work thus traveled with him around the world, and my remains to be done in giving shape to Mackey’s mother and I left him working on countless park dreams of a group-theoretic universe. benches while we pursued separate adventures. In recent years, when attention has been giv- My father was notorious- en to quantum systems arising from models of ly eccentric and proud of space-time based on non-Archimedean geome- it. Having found a clothing try, or super geometry, the Mackey formulation style that suited him early has offered the surest guide to progress. As an in life, gold pocket watch in- example one may mention the classification of su- cluded, he resisted straying perparticles carried out in the 1970s by physicists any further from that than whose rigorous formulation needs an extension absolutely required. No mat- of Mackey’s imprimitivity theorem to the context ter how hot it got, he could of homogeneous spaces for the super Lie groups, rarely be persuaded to re- such as the super Poincaré groups. move his jacket and tie, even In all my encounters with him he was always at the beach (and we have considerate, full of humor, and never conde- the pictures to prove it here scending. He was aware that his strength was today). I will leave to your more in ideas than in technique (although his imaginations the discussions best work reveals technical mastery in functional Photo courtesy of Ann Mackey. he had with my mother on the analysis of a high order) and repeatedly told me subject of his attire. Mackey with daughter Ann that as one gets older, technique will disappear He disclaimed any formal in Zurich, 1971. and so one has to pay more attention to ideas. His interest in people, but his view of mathematics and its role in the physical conversations were filled with news of those he world was a mature one, full of understanding had encountered during his day. Although he re- and admiration for the physicists’ struggles to sisted any event that might interfere with his create a coherent world picture, yet aware of the work time, he actively enjoyed socializing. Bed- important but perhaps not decisive role of mathe- time was sacred, though, so he was often the first matics in its creation. In his own way he achieved to leave a party. a beautiful synthesis of mathematics and physics The visual arts left him unmoved; instead, he that will be the standard for many years to come. saw great beauty in mathematics. New insights thrilled him, and he would emerge from his study Ann Mackey giddy with excitement about a new idea. He was not musical, but enjoyed classical mu- Eulogy for My Father, George Mackey* sic in modest doses and had a great fondness for Before I lose my composure, I want to offer Gilbert and Sullivan tunes. He would often burst thanks. My mother and I thank my father for into song spontaneously, generally off-key. He everything he was to us, and we thank everyone taught himself to play the piano by creating his here today for coming to share in this memorial own numeric notational system, with pluses for service, especially those who have spoken and sharps and minus signs for flats. He called it the captured my father’s essence so beautifully. “touch typewriter method” and built up quite a My father had a wonderful life and knew repertoire of his favorite tunes. He took a similar it. Having started out feeling like a misfit who approach to foreign language study. He was more sentimental than he would ever *Remarks at the memorial service, Harvard Chapel, April admit, often choking up as he read certain pas- 29, 2006. sages aloud.

August 2007 Notices of the AMS 849 Many people have already touched on my father’s honesty and adherence to principle, so I’m going try to avoid redundancy. However, I did want to stress that as much as he enjoyed a good argument and as much as his honesty could sometimes come at the expense of tact, he never wished to be unkind. He often agonized about how to deliver his message without giving serious offense. He could be absolutely infuriating, and there was sometimes an element of mischief and contrariness in his arguments, but his words truly bore no malice. He rarely signed on to any idea Photo courtesy of Ann Mackey. or behavior before he had thoroughly researched the subject in George Mackey. question and had reached his own And although mystified that anyone, including my- conclusions. He was not a quick self, would not choose a career in the sciences, he study, and this could be infuriating believed that one should follow one’s heart and al- when something seemed obvious on ways supported the choices I made, regardless of its face. We joked that despite his whetherhefullyunderstoodthem. physical caution, his aversion to tak- He was a dedicated letter writer, sending long, ing orders might someday cause affectionate, newsy updates to us when he trav- him to walk straight off a precipice eled, and for me, scientific and mathematical ex- if he wasn’t given clear, provable positions, including letters that would end, and evidence that the precipice existed. I quote, “Q.E.D. Love, Daddy.” In his later years I loved my father dearly. Al- he eagerly embraced email as a way to communi- though he was fond of asserting cate with me. At my request, he sent me a serial- to anyone who would listen that ized account of his life, filled with all the stories he’d never wanted a family, it was he’d told me of his childhood and continuing part- clear to everyone that once he stum- way through graduate school. A year or so ago, Photo courtesy of Robert Doran. bled into marriage and fatherhood, although he had lost the ability to put more than a Mackey with his he relished and cherished it, even few words on paper, he narrated further accounts famous clipboard, as he struggled to adapt to the aloud for my mother to transcribe. We have all 1988. compromises it asked of him. these words to hold on to, and I am deeply grate- His marriage to my mother, his ful for that. opposite in so many ways, had its share of mem- Please forgive me if I take this moment to en- orably dramatic moments, but at its core was courage everyone here to make the time to write love and respect. They lived much of their daily to those they love. lives independently, but were together for most My father was a realist about death. Conscious meals, especially their nightly cocktails and can- of being an older parent, he had, to some extent, dlelit dinners, where they shared news of their been preparing me for his eventual demise virtu- days, followed by my father reading aloud to my ally since the day I was born. But it was very hard mother as I drifted off to asleep upstairs. They to watch the man we knew and loved slip away enriched each other’s lives in ways neither expect- from us over these past few years. We are fortu- ed and particularly enjoyed their travels together nate that in many fundamental ways he did retain and the friends they made around the globe. De- much of his old self, but it was a sad journey. spite their different outlooks on many things, Certainly we appreciate that death is part of each was the other’s best and most trusted friend. the natural cycle of life and that we should rejoice As serious as my father was about his work, that he lived so fully for so long. Those thoughts, he was a child at heart. He was a wonderful, play- and the memories he left, and those shared by so ful father to me when I was young, reading to me, many people in their letters and phone calls do drawing me pictures, crawling around on all fours, help immensely. I am glad that his grandchildren and patiently playing endless games of Monopoly, were at least able to spend time with him, if not know him as he once was. But standing right here, so long as we kept the latter to morning play times, I am conscious mostly of what we have lost. My lestthecompetitivestressofthegameinterferewith mother and I miss him terribly, and it is very hard his sleep. He was a thoughtful, concerned—albeit to say goodbye. somewhat befuddled—advocate for me as I navi- gated the tricky landscape of adolescence. He was a refuge when I was sad or anxious. He was patient.

850 Notices of the AMS Volume 54, Number 7