Equivariance and Imprimitivity for Discrete Hopf C*-Coactions

TL;DR

This paper explores the structure of coactions of discrete Hopf C*-algebras, establishing isomorphisms between tensor products of imprimitivity bimodules and crossed products, advancing the understanding of equivariance and imprimitivity in this setting.

Contribution

It introduces new notions for bimodules of coactions, defines their restrictions and duals, and proves an isomorphism linking these structures, providing a natural transformation between crossed-product functors.

Findings

Established an isomorphism between tensor product bimodules involving imprimitivity bimodules and crossed products.

Defined notions for bimodules of coactions and their restrictions, duals, and crossed products.

Provided a framework for understanding equivariance and imprimitivity in discrete Hopf C*-coactions.

Abstract

Let U, V, and W be multiplicative unitaries coming from discrete Kac systems such that W is an amenable normal submultiplicative unitary of V with quotient U. We define notions for right-Hilbert bimodules of coactions of S_V and (S_V)^, their restrictions to S_W and (S_U)^, their dual coactions, and their full and reduced crossed products. If N(A) denotes the imprimitivity bimodule associated to any coaction of S_V on a C*-algebra A by Ng's imprimitivity theorem, then for any suitably nondegenerate injective coaction of S_V on a right-Hilbert A - B bimodule X we establish an isomorphism between two tensor product bimodules involving N(A), N(B), and certain crossed products of X. This can be interpreted as a natural transformation between two crossed-product functors.