George Whitelaw Mackey (February 1, 1916 – March 15, 2006) was an American mathematician known for his contributions to quantum logic, representation theory, and noncommutative geometry.

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 universitie ...

(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 ...

at Harvard University 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 ve ...

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 Hi ...

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 algebras and noncommutative geometry. A central role in Mackey's work, both in the theory of group representations and in mathematical physics, was played by the concepts of system of imprimitivity and induced representations. 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 w ...

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 groups using the method of orbits developed by

Alexandre Kirillov Alexandre Aleksandrovich Kirillov (russian: Алекса́ндр Алекса́ндрович Кири́ллов, born 1936) is a Soviet and Russian mathematician, known for his works in the fields of representation theory, topological groups a ...

in the 1960s. His notion of "virtual subgroup", introduced in 1966 using the language of groupoids, 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 of a locally compact group (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 ...

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 continuou ...

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

statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...

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 (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, and ...

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 Lawrence G. Brown (born February 6, 1943 in St. Louis, Missouri) is an American mathematician who studies operator algebras. Brown studied at Harvard University, graduating in 1968 with George Mackey as his advisor and thesis entitled ''On the Str ...

, Paul Chernoff, Edward G. Effros, Calvin Moore, Richard Palais,

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 24 March 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 Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...

'' 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