Marc Aristide Rieffel is a mathematician noted for his fundamental contributions to

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 ...

G Cortinas (2008) ''K-theory and Noncommutative Geometry'',

European Mathematical Society The European Mathematical Society (EMS) is a European organization dedicated to the development of mathematics in Europe. Its members are different mathematical societies in Europe, academic institutions and individual mathematicians. The curren ...

. and

quantum group In mathematics and theoretical physics, the term quantum group denotes one of a few different kinds of noncommutative algebras with additional structure. These include Drinfeld–Jimbo type quantum groups (which are quasitriangular Hopf algebra ...

theory. He is currently a professor in the department of mathematics at the

University of California, Berkeley The University of California, Berkeley (UC Berkeley, Berkeley, Cal, or California), is a Public university, public Land-grant university, land-grant research university in Berkeley, California, United States. Founded in 1868 and named after t ...

. In 2012, he was selected as one of the inaugural fellows of the

American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...

.List of Fellows of the American Mathematical Society
retrieved 2014-03-17.

Contributions

Rieffel earned his doctorate from

Columbia University Columbia University in the City of New York, commonly referred to as Columbia University, is a Private university, private Ivy League research university in New York City. Established in 1754 as King's College on the grounds of Trinity Churc ...

in 1963 under

Richard Kadison Richard Vincent Kadison (July 25, 1925 – August 22, 2018) was an American mathematician known for his contributions to the study of operator algebras. Career Born in New York City in 1925, Kadison was a Gustave C. Kuemmerle Professor in the De ...

with a dissertation entitled ''A Characterization of Commutative Group Algebras and Measure Algebras''. Rieffel introduced

Morita equivalence In abstract algebra, Morita equivalence is a relationship defined between rings that preserves many ring-theoretic properties. More precisely, two rings ''R'', ''S'' are Morita equivalent (denoted by R\approx S) if their categories of modules ar ...

as a fundamental notion in

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 ...

and as a tool for classifying C*-algebras. For example, in 1981 he showed that if ''A''_''θ'' denotes the

noncommutative torus In mathematics, and more specifically in the theory of C*-algebras, the noncommutative tori ''A''θ, also known as irrational rotation algebras for irrational values of θ, form a family of noncommutative C*-algebras which generalize the algebra ...

of angle ''θ'', then ''A''_''θ'' and ''A''_''η'' are Morita equivalent if and only if ''θ'' and ''η'' lie in the same orbit of the action of SL(2, Z) on R by

fractional linear transformations In mathematics, a linear fractional transformation is, roughly speaking, an invertible transformation of the form : z \mapsto \frac . The precise definition depends on the nature of , and . In other words, a linear fractional transformation is a ...

. More recently, Rieffel has introduced a noncommutative analogue of Gromov-Hausdorff convergence for

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...

metric spaces In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are a general setting for ...

which is motivated by applications to

string theory In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and intera ...

References

External links

* {{DEFAULTSORT:Rieffel, Marc Living people 20th-century American mathematicians 21st-century American mathematicians Columbia Graduate School of Arts and Sciences alumni Fellows of the American Mathematical Society University of California, Berkeley faculty 1937 births