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 continuous ...

G Cortinas (2008) ''K-theory and Noncommutative Geometry'', European Mathematical Society. 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 algebras) ...

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 land-grant research university in Berkeley, California. Established in 1868 as the University of California, it is the state's first land-grant u ...

. 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 (also known as Columbia, and officially as Columbia University in the City of New York) is a private research university in New York City. Established in 1754 as King's College on the grounds of Trinity Church in Manhatt ...

in 1963 under

Richard Kadison Richard Vincent Kadison (July 25, 1925 – August 22, 2018)F ...

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 like ''R'', ''S'' are Morita equivalent (denoted by R\approx S) if their categories of modules ...

as a fundamental notion in noncommutative geometry 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, a transformation of the form :z \mapsto \frac , which has an inverse. The precise definition depends on the nature of , and . In other words, a linear fractional transfor ...

. 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 * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

metric spaces 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 interac ...

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