George Whitelaw Mackey (February 1, 1916 – March 15, 2006) was an American mathematician known for his contributions to

quantum logic In the mathematical study of logic and the physical analysis of quantum foundations, quantum logic is a set of rules for manipulation of propositions inspired by the structure of quantum theory. The formal system takes as its starting p ...

representation theory Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...

, and

noncommutative geometry Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of ''spaces'' that are locally presented by noncommutative algebras of functions, possibly in some g ...

Career

Mackey earned his B.A. at

Rice University William Marsh Rice University, commonly referred to as Rice University, is a Private university, private research university in Houston, Houston, Texas, United States. Established in 1912, the university spans 300 acres. Rice University comp ...

in 1938 and obtained his Ph.D. at

Harvard University Harvard University is a Private university, private Ivy League research university in Cambridge, Massachusetts, United States. Founded in 1636 and named for its first benefactor, the History of the Puritans in North America, Puritan clergyma ...

in 1942 under the direction of Marshall H. Stone. He joined the Harvard University Mathematics Department in 1943, was appointed Landon T. Clay Professor of Mathematics and Theoretical Science in 1969 and remained there until he retired in 1985.

Work

Earlier in his career Mackey did significant work in the duality theory of

locally convex space In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vec ...

s, which provided tools for subsequent work in this area, including

Alexander Grothendieck Alexander Grothendieck, later Alexandre Grothendieck in French (; ; ; 28 March 1928 – 13 November 2014), was a German-born French mathematician who became the leading figure in the creation of modern algebraic geometry. His research ext ...

's work on topological tensor products. Mackey was one of the pioneer workers in the intersection of

, the theory of

infinite-dimensional In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to d ...

unitary representation In mathematics, a unitary representation of a group ''G'' is a linear representation π of ''G'' on a complex Hilbert space ''V'' such that π(''g'') is a unitary operator for every ''g'' ∈ ''G''. The general theory is well-developed in the ca ...

s of

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...

s, the theory of

operator algebra In functional analysis, a branch of mathematics, an operator algebra is an algebra of continuous linear operators on a topological vector space, with the multiplication given by the composition of mappings. The results obtained in the study o ...

s and

. A central role in Mackey's work, both in the theory of group representations and in

mathematical physics Mathematical physics is the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the de ...

, was played by the concepts of system of imprimitivity and

induced representation In group theory, the induced representation is a group representation, representation of a group, , which is constructed using a known representation of a subgroup . Given a representation of '','' the induced representation is, in a sense, the "m ...

s. This idea led naturally to an analysis of the representation theory of semi-direct products in terms of ergodic actions of groups and in some cases a complete classification of such representations. Mackey's results were essential tools in the study of the representation theory of nilpotent Lie groups using the method of orbits developed by Alexandre Kirillov in the 1960s. His notion of "virtual subgroup", introduced in 1966 using the language of

groupoid In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a: * '' Group'' with a partial fu ...

s, had a significant influence in

ergodic theory Ergodic theory is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, "statistical properties" refers to properties which are expressed through the behav ...

. Another essential ingredient in Mackey's work was the assignment of a Borel structure to the

dual object In category theory, a branch of mathematics, a dual object is an analogue of a dual vector space from linear algebra for Object (category theory), objects in arbitrary Monoidal category, monoidal categories. It is only a partial generalization, base ...

of a

locally compact group In mathematics, a locally compact group is a topological group ''G'' for which the underlying topology is locally compact and Hausdorff. Locally compact groups are important because many examples of groups that arise throughout mathematics are lo ...

(specifically a locally compact separable metric group) ''G''. One of Mackey's important conjectures, which was eventually solved by work of James Glimm on

C*-algebra In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of contin ...

s, was that ''G'' is type I (meaning that all its factor representations are of type I) if and only if the Borel structure of its dual is a

standard Borel space In mathematics, a standard Borel space is the Borel space associated with a Polish space. Except in the case of discrete Polish spaces, the standard Borel space is unique, up to isomorphism of measurable spaces. Formal definition A measurable ...

. He has written numerous survey articles connecting his research interests with a large body of mathematics and physics, particularly

quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...

and

statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...

Honours and students

Mackey was among the first five recipients of William Lowell Putnam fellowships in 1938. He received the Leroy P. Steele Prize in 1975 for his article ''Ergodic theory and its significance for statistical mechanics and probability theory''. Mackey was an elected member of the

American Academy of Arts and Sciences The American Academy of Arts and Sciences (The Academy) is one of the oldest learned societies in the United States. It was founded in 1780 during the American Revolution by John Adams, John Hancock, James Bowdoin, Andrew Oliver, and other ...

the

National Academy of Sciences The National Academy of Sciences (NAS) is a United States nonprofit, NGO, non-governmental organization. NAS is part of the National Academies of Sciences, Engineering, and Medicine, along with the National Academy of Engineering (NAE) and the ...

and the

American Philosophical Society The American Philosophical Society (APS) is an American scholarly organization and learned society founded in 1743 in Philadelphia that promotes knowledge in the humanities and natural sciences through research, professional meetings, publicat ...

. Lawrence G. Brown, Paul Chernoff, Edward G. Effros, Calvin Moore,

Richard Palais Richard Sheldon Palais (born May 22, 1931) is an American mathematician working in differential geometry. Education and career Palais studied at Harvard University, where he obtained a B.A. in 1952, an M.A. in 1954 and a Ph.D. in 1956. His Ph ...

Caroline Series Caroline Mary Series (born 24 March 1951) is an English mathematician known for her work in hyperbolic geometry, Kleinian groups and dynamical systems. Early life and education Series was born on March 24, 1951, in Oxford to Annette and Georg ...

, John Wermer and Robert Zimmer have been doctoral students of Mackey.

Andrew Gleason Andrew Mattei Gleason (19212008) was an American mathematician who made fundamental contributions to widely varied areas of mathematics, including the solution of Hilbert's fifth problem, and was a leader in reform and innovation in teaching at ...

had no PhD, but considered Mackey to be his advisor.

Books

*''Mathematical Foundations of Quantum Mechanics'' (Dover Books on Mathematics, 2004 ) *''Unitary Group Representations in Physics, Probability, and Number Theory'', 402 pages, Benjamin–Cummings Publishing Company (1978), *''The Theory of Unitary Group Representations'' (Chicago Lectures in Mathematics) University Of Chicago Press (August 1, 1976) *''Induced representations of groups and quantum mechanics'', Publisher: W. A. Benjamin (1968) *''Mathematical Problems of Relativistic Physics'' (Lectures in Applied Mathematics Series, Vol 2) by I. E. Segal, George Whitelaw Mackey, Publisher: Amer Mathematical Society (June 1967) *''Lectures on the theory of functions of a complex variable'' Publisher: R. E. Krieger Pub. Co (1977)

References

External links

* *
George Mackey (1916–2006)
''

Notices of the American Mathematical Society ''Notices of the American Mathematical Society'' is the membership journal of the American Mathematical Society (AMS), published monthly except for the combined June/July issue. The first volume was published in 1953. Each issue of the magazine ...

''; vol. 54, no. 7 (August 2007).
George Mackey (1 February 1916–15 March 2006)
''Proceedings of the American Philosophical Society''; vol. 152, no. 4 (December 2008).

*http://archive.boston.com/news/globe/obituaries/articles/2006/04/28/george_mackey_professor_devoted_to_truth_theorems/?page=1 Obituary from Boston Globe]
Peter Woit's blog entry on Mackey
*Two letters from George Mackey and the text of his speech "What do Mathematicians Do?, collected by Stephanie Singer
First letter

Second letter

Speech
{{DEFAULTSORT:Mackey, George Whitelaw 1916 births Members of the United States National Academy of Sciences 2006 deaths 20th-century American mathematicians 21st-century American mathematicians American mathematical analysts American topologists Putnam Fellows Rice University alumni Harvard University alumni Harvard University Department of Mathematics faculty People from St. Louis Mathematicians from Missouri Functional analysts