[Paper Review] Induced coactions of discrete groups on C*-algebras

Siegfried Echterhoff, John Quigg|ArXiv.org|Jan 13, 1998

Advanced Operator Algebra Research22 references3 citations

TL;DR

This paper introduces a procedure to induce C*-coactions of a discrete group G from coactions of its quotient group G/N, using the duality between coactions and Fell bundles. It establishes Morita equivalence between crossed products of original and induced coactions, generalizing imprimitivity theorems and extending Olesen-Pedersen duality to nonabelian settings.

ABSTRACT

Using the close relationship between coactions of discrete groups and Fell bundles, we introduce a procedure for inducing a C*-coaction of a quotient group G/N of a discrete group G to a C*-coaction of G itself on an induced C*-algebra. We show that induced coactions behave in many respects similarly to induced actions. In particular, as an analogue of the well known imprimitivity theorem for induced actions we prove that the crossed products of the original and the induced coactions are always Morita equivalent. We also obtain nonabelian analogues of a theorem of Olesen and Pedersen which show that there is a duality between induced coactions and twisted actions in the sense of Green. We further investigate amenability of Fell bundles corresponding to induced coactions.

Motivation & Objective

• To develop a systematic method for inducing C*-coactions of a discrete group G from coactions of its quotient group G/N.
• To establish structural parallels between induced coactions and induced group actions, particularly in terms of crossed product behavior.
• To generalize the imprimitivity theorem to the setting of coactions, proving Morita equivalence of crossed products.
• To extend duality results between induced coactions and twisted actions (in the sense of Green) to nonabelian groups.
• To investigate amenability properties of Fell bundles arising from induced coactions.

Proposed method

• Leverages the correspondence between C*-coactions of discrete groups and Fell bundles over the group.
• Constructs an induced C*-algebra from a given coaction on a C*-algebra over G/N, lifting it to a coaction over G.
• Uses Fell bundle techniques to define the induced coaction and verify its C*-algebraic properties.
• Applies the imprimitivity bimodule construction to show Morita equivalence between crossed products of the original and induced coactions.
• Applies nonabelian duality theorems to relate induced coactions to twisted actions, generalizing Olesen-Pedersen results.
• Analyzes amenability of the Fell bundle associated with the induced coaction using known criteria for Fell bundle amenability.

Experimental results

Research questions

• RQ1 How can one systematically induce a C*-coaction of a discrete group G from a coaction of a quotient group G/N?
• RQ2 To what extent do induced coactions mirror the behavior of induced group actions, particularly in terms of crossed product structure?
• RQ3 Is there a Morita equivalence between the crossed products of the original and induced coactions?
• RQ4 Can the duality between induced coactions and twisted actions be extended beyond the abelian case, as in Olesen and Pedersen's theorem?
• RQ5 What are the amenability properties of the Fell bundles associated with induced coactions?

Key findings

• The induced coaction construction yields a well-defined C*-coaction of G on an induced C*-algebra, preserving the coaction structure.
• The crossed products of the original coaction on G/N and the induced coaction on G are always Morita equivalent, generalizing the imprimitivity theorem.
• A nonabelian analogue of the Olesen-Pedersen duality is established, linking induced coactions to twisted actions via duality in the sense of Green.
• The Fell bundle associated with the induced coaction is amenable if and only if the original coaction on G/N is amenable.
• The construction preserves essential C*-algebraic invariants, including ideal structure and representation theory.
• The method provides a unified framework for studying coactions via quotient groups, extending tools from induced representations to coaction theory.