A categorical approach to imprimitivity theorems for 𝐶*-dynamical systems

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. RIGHT-HILBERT BIMODULES AND PARTIAL IMPRIMITIVITY BIMODULES *Cc(G,M β (B))

Group actions on topological graphs

“…For the proof of Theorem 5.6 below, we will need to make extensive use of the multiplier bimodules of [EKQR06]; the relevant facts are collected in Appendix A.…”

“…To prepare for the general framework introduced below, we begin with some prefatory comments. First, to establish context, we recall that the "C-multipliers" in [EKQR06] involved tensor products: if A and C are C * -algebras, then [EKQR06] defined M C (A⊗C) to be all multipliers of A⊗C that multiply 1⊗C into A⊗C. Here we need a generalization to situations where there are no tensor products.…”

Naturality of Rieffel’s Morita Equivalence for Proper Actions

“…It follows from [3,Theorem 3.7] that the assignments (A, α) → A ⋊ α,r G and (X, u) → X ⋊ u,r G give a functor from C*act(G) to C* (since C*act(G) and C* are subcategories of the categories considered in [3,Theorem 3.7]). After restricting to the semi-comma category we still have a functor.…”

“…The objects in our underlying category C* are C * -algebras, and the morphisms from A to B are isomorphism classes of the rightHilbert A -B bimodules which were invented by Rieffel [35] to place the theory of induced representations of groups in a C * -algebraic setting. The category C* was introduced in [2] and [3] as a setting for imprimitivity theorems for crossed products by actions and coactions, and was independently discovered in other contexts by Landsman [18,19] and by Schweizer [38]. It has the attractive feature that the invertible morphisms are those which are based on imprimitivity bimodules (see, for example, [2, Proposition 2.6]).…”

Pseudo-multiplicative unitaries on C*-modules and Hopf C*-families I

“…If we want to emphasize that the factor F of a (flipped) internal tensor product E F (or F E) is considered as a right C*-bimodule via a fixed representation , we denote the product by E F (or F E, respectively). We shall frequently use the following result [3], Proposition 1.34: S T 2 L B .E 1 F 1 ; E 2 F 2 / such that .S T /.Á / D SÁ T for all Á 2 E 1 , 2 F 1 . Moreover, .S T / D S T .…”