[Paper Review] A Categorical Approach to Imprimitivity Theorems for C*-Dynamical Systems

Siegfried Echterhoff, S. Kaliszewski+2 more|ArXiv.org|May 30, 2002

Advanced Operator Algebra Research40 references48 citations

TL;DR

This paper presents a unified categorical framework for understanding key imprimitivity theorems in C*-dynamical systems by treating them as natural equivalences between crossed-product functors. It uses C*-algebras with actions or coactions of a fixed locally compact group G as objects, and equivariant equivalence classes of right-Hilbert bimodules as morphisms, with composition via balanced tensor products, revealing deep structural connections between Green's, Mansfield's, and induced algebra imprimitivity theorems.

ABSTRACT

Imprimitivity theorems provide a fundamental tool for studying the representation theory and structure of crossed-product C*-algebras. In this work, we show that the Imprimitivity Theorem for induced algebras, Green's Imprimitivity Theorem for actions of groups, and Mansfield's Imprimitivity Theorem for coactions of groups can all be viewed as natural equivalences between various crossed-product functors among certain equivariant categories. The categories involved have C*-algebras with actions or coactions (or both) of a fixed locally compact group G as their objects, and equivariant equivalence classes of right-Hilbert bimodules as their morphisms. Composition is given by the balanced tensor product of bimodules. The functors involved arise from taking crossed products; restricting, inflating, and decomposing actions and coactions; inducing actions; and various combinations of these. Several applications of this categorical approach are also presented, including some intriguing relationships between the Green and Mansfield bimodules, and between restriction and induction of representations.

Motivation & Objective

• To unify major imprimitivity theorems in C*-dynamical systems under a single categorical framework.
• To clarify the structural relationships between crossed-product functors arising from actions, coactions, and their restrictions, inductions, and decompositions.
• To provide a conceptual foundation for understanding the equivalence of representations and bimodules in duality and induction theory.
• To reveal hidden symmetries and dualities between Green's and Mansfield's bimodules through categorical equivalence.

Proposed method

• Modeling C*-dynamical systems as objects in a category with actions or coactions of a fixed locally compact group G.
• Using equivariant equivalence classes of right-Hilbert bimodules as morphisms, with composition defined by the balanced tensor product.
• Defining functors from the category of G-C*-algebras to crossed-product categories via restriction, inflation, decomposition, induction, and crossed product construction.
• Establishing natural isomorphisms between these functors to realize the Imprimitivity Theorems as categorical equivalences.
• Applying the categorical framework to derive structural results about representations and bimodules in duality and induction.
• Demonstrating that the Green and Mansfield bimodules arise as dual constructions under this framework.

Experimental results

Research questions

• RQ1 How can the Imprimitivity Theorem for induced algebras be expressed as a natural equivalence in a categorical setting?
• RQ2 What categorical structure underlies Green’s Imprimitivity Theorem for group actions on C*-algebras?
• RQ3 How does Mansfield’s Imprimitivity Theorem for coactions relate to the categorical framework of crossed-product functors?
• RQ4 What is the relationship between the Green and Mansfield bimodules in terms of duality and equivalence in the category of G-C*-algebras?
• RQ5 Can restriction and induction of representations be systematically described using the categorical equivalence of crossed-product functors?

Key findings

• The Imprimitivity Theorem for induced algebras is realized as a natural equivalence between a crossed-product functor and a restriction functor in the categorical framework.
• Green’s Imprimitivity Theorem is shown to be a natural isomorphism between functors involving induced actions and crossed products with the dual group.
• Mansfield’s Imprimitivity Theorem is interpreted as a categorical equivalence between functors arising from coactions and their reduced crossed products.
• The Green and Mansfield bimodules are identified as dual constructions under the categorical equivalence, revealing a deep symmetry between the two theorems.
• Restriction and induction of representations are systematically described as functors that preserve the categorical equivalences established by the imprimitivity theorems.
• The framework provides a conceptual unification of duality, induction, and restriction in C*-dynamical systems through the language of functors and natural isomorphisms.