imprimitivity

Imprimitivity is a concept in mathematics, particularly in representation theory and operator algebras, describing a compatibility structure between a group action or transformation group and representations.

In the theory of group representations, a system of imprimitivity for a locally compact group G acting on a measure space X consists of a unitary representation U of G on a Hilbert space H together with a projection-valued measure E on X on H that satisfies covariance: U_g E(S) U_g^* = E(g S) for all g in G and measurable S subset X. Such systems encode how the group action interacts with observables on X and are central to the theory of induced representations. Mackey's imprimitivity theorem characterizes irreducible representations that admit a transitive system of imprimitivity: every such representation is induced from a representation of a stabilizer subgroup, yielding a correspondence between representations of G and representations of subgroups of the form G_x.

In operator algebra theory, imprimitivity also denotes structures implementing Morita equivalences between crossed product C*-algebras. Rieffel's imprimitivity theorem provides conditions under which two transformation group C*-algebras are Morita equivalent, mediated by an imprimitivity bimodule carrying a compatible action. This connects dynamical systems to noncommutative geometry and strengthens tools for analyzing representations and crossed products.

Historically, the term originated in Mackey’s work on induced representations and systems of imprimitivity, with a C*-algebraic analogue developed by Rieffel in the 1970s.