George Whitelaw Mackey (February 1, 1916 – March 15, 2006) was an American

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

known for his contributions to quantum logic,

representation theory Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...

, 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 bachelor of arts at

Rice University William Marsh Rice University (Rice University) is a private research university in Houston, Texas. It is on a 300-acre campus near the Houston Museum District and adjacent to the Texas Medical Center. Rice is ranked among the top universities ...

(then the Rice Institute) in 1938 and obtained his

Ph.D. A Doctor of Philosophy (PhD, Ph.D., or DPhil; Latin: or ') is the most common degree at the highest academic level awarded following a course of study. PhDs are awarded for programs across the whole breadth of academic fields. Because it is ...

Harvard University Harvard University is a private Ivy League research university in Cambridge, Massachusetts. Founded in 1636 as Harvard College and named for its first benefactor, the Puritan clergyman John Harvard, it is the oldest institution of high ...

in 1942 under the direction of

Marshall H. Stone Marshall Harvey Stone (April 8, 1903 – January 9, 1989) was an American mathematician who contributed to real analysis, functional analysis, topology and the study of Boolean algebras. Biography Stone was the son of Harlan Fiske Stone, who wa ...

. 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 v ...

s, which provided tools for subsequent work in this area, including Alexander Grothendieck's work on

topological tensor product In mathematics, there are usually many different ways to construct a topological tensor product of two topological vector spaces. For Hilbert spaces or nuclear spaces there is a simple well-behaved theory of tensor products (see Tensor product of ...

s. Mackey was one of the pioneer workers in the intersection of quantum logic, the theory of infinite-dimensional

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 case ''G ...

s of groups, 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 of ...

s and

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

mathematical physics Mathematical physics refers to the development of 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 developme ...

, was played by the concepts of system of imprimitivity and

induced representation In group theory, the induced representation is a 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 "most general" represe ...

s. This idea led naturally to an analysis of the representation theory of

semi-direct product In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product: * an ''inner'' semidirect product is a particular way in ...

s 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 group In mathematics, specifically group theory, a nilpotent group ''G'' is a group that has an upper central series that terminates with ''G''. Equivalently, its central series is of finite length or its lower central series terminates with . Intuit ...

s 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 func ...

s, had a significant influence in ergodic theory. 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 objects in arbitrary monoidal categories. It is only a partial generalization, based upon the categorical properties of dual ...

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 loc ...

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

James Glimm James Gilbert Glimm (born March 24, 1934) is an American mathematician, former president of the American Mathematical Society, and distinguished professor at Stony Brook University. He has made many contributions in the areas of pure and applie ...

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

quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistr ...

and statistical mechanics.

Honours and students

Mackey was among the first five recipients of

William Lowell Putnam William Lowell Putnam II (November 22, 1861 – June 1923) (more commonly known as William Putnam, Sr.) was an American lawyer and banker. Putnam was the son of George and Harriet (Lowell) Putnam. He graduated from Harvard in 1882, and proc ...

fellowships in 1938. He received the

Leroy P. Steele Prize The Leroy P. Steele Prizes are awarded every year by the American Mathematical Society, for distinguished research work and writing in the field of mathematics. Since 1993, there has been a formal division into three categories. The prizes have b ...

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 (abbreviation: AAA&S) 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, a ...

the National Academy of Sciences and the

American Philosophical Society The American Philosophical Society (APS), founded in 1743 in Philadelphia, is a scholarly organization that promotes knowledge in the sciences and humanities through research, professional meetings, publications, library resources, and communit ...

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

Richard Palais Richard Sheldon Palais (born May 22, 1931) is a mathematician working in geometry who introduced the principle of symmetric criticality, the Mostow–Palais theorem, the Lie–Palais theorem, the Morse–Palais lemma, and the Palais–Smale c ...

, Caroline Series, 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''; 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).
Obituary from Boston GlobePeter 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 Mathematical analysts Topologists Putnam Fellows Rice University alumni Harvard University alumni Harvard University faculty People from St. Louis Mathematicians from Missouri