Rieffel

In mathematics, "Rieffel" commonly refers to aspects of the work of Marc Rieffel, particularly in the areas of operator algebras and quantum groups. Here's a breakdown of common usages:

Rieffel Induction: This is a method, developed by Marc Rieffel, for constructing representations of a C*-algebra from representations of a C*-subalgebra. It provides a way to induce representations from a subgroup to the larger group in the context of C*-algebras and their representations. The induction process involves forming a Hilbert module and using it to define a representation of the larger algebra.

Rieffel Deformation Quantization: This refers to a method of deforming the product on a commutative algebra of functions on a manifold, typically phase space, to obtain a non-commutative algebra that can be interpreted as a quantization of the original classical system. Rieffel deformation quantization uses techniques from C*-algebra theory and involves twisting the multiplication using a 2-cocycle related to a Poisson structure on the manifold. It provides a rigorous mathematical framework for quantization.

Rieffel's Pseudodifferential Calculus: Rieffel developed a pseudodifferential calculus on quantum tori. This extends the traditional pseudodifferential calculus on Euclidean space to the noncommutative setting of quantum tori, allowing for the study of operators and their properties in this context.

Rieffel Modules/Hilbert Modules: These are modules over a C*-algebra that are equipped with a "C*-valued inner product". These modules play a crucial role in Rieffel induction and in the theory of Morita equivalence of C*-algebras. They generalize the notion of Hilbert spaces.

In summary, the name "Rieffel" is associated with significant contributions to the theory of operator algebras, particularly in the areas of representation theory, deformation quantization, and noncommutative geometry. His work provides tools and techniques for studying noncommutative structures and their connections to classical mathematics.