A structure theorem in representation theory of locally compact groups that characterizes unitary representations induced from a closed subgroup in terms of systems of imprimitivity (projection-valued measures) on the homogeneous space and yields an equivalence between certain categories of representations; it relates induced representations to covariant representations of C*-dynamical systems arising from the homogeneous space.

Principle

Induction of unitary representations can be encoded by a compatible projection-valued measure (a system of imprimitivity) on the homogeneous space G/H; the organizing idea is that transitive G-spaces and their projection-valued measures classify induced unitary representations up to equivalence.

Demonstration

For G locally compact and H closed, the representation induced from a unitary representation σ of H acts by left translations on sections of a Hilbert bundle over G/H; the theorem identifies this induced representation with a representation of G together with a projection-valued measure on G/H satisfying covariance relations, e.g., the regular representation on L^2(G/H) arises from the trivial representation of H.

Misapplication

Applying the theorem outside its hypotheses (non-locally-compact groups, nonunitary representations, or neglecting the measurable structure and modular function) or treating algebraic induction without addressing the analytic aspects (measures, integrability).

Consequence

Gives a powerful classification tool: induced unitary representations are described by geometric data on homogeneous spaces, which facilitates decomposition, imprimitivity-based equivalences of categories, and connections to crossed-product C*-algebras and Mackey’s machine in ergodic theory and noncommutative harmonic analysis.

Reversal

The absence of a system of imprimitivity that satisfies the covariance prevents realizing a representation as induced from a subgroup; equivalently, not every unitary representation is induced from a proper subgroup—induction is a restricted construction.

Boundary

Requires locally compact groups, closed subgroups, unitary representations on Hilbert spaces, and measurable/projective structures; it does not directly apply to purely algebraic induction, to nonunitary Banach space representations, or to groups lacking a reasonable Haar measure.

Semantic Tension

Relates to Frobenius reciprocity and to the theory of induced representations in purely algebraic settings; the tension is between analytic (measurable, unitary) induction captured by imprimitivity and algebraic induction where measure-theoretic covariance is absent.

Synthesis

The Mackey imprimitivity theorem states that unitary representations induced from a closed subgroup are exactly those that admit a compatible projection-valued measure on the homogeneous space; it packages induction as geometric data on G/H and yields categorical equivalences useful in decomposition and crossed-product theory.