Skew products and crossed products by coactions

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15 Scopus citations

Abstract

Given a labeling c of the edges of a directed graph E by elements of a discrete group G, one can form a skew-product graph E ×c G. We show, using the universal properties of the various constructions involved, that there is a coaction δ of G on C* (E) such that C* (E ×c G) is isomorphic to the crossed product C* (E) ×δ G. This isomorphism is equivariant for the dual action δ̂ and a natural action γ of G on C* (E ×c G); following results of Kumjian and Pask, we show that C* (E ×c G) ×γ G ≅ C* (E ×c G) ×γ,r G ≅ C* (E) ⊗ K(ℓ2(G)), and it turns out that the action γ is always amenable. We also obtain corresponding results for r-discrete groupoids Q and continuous homomorphisms c : Q → G, provided Q is amenable. Some of these hold under a more general technical condition which obtains whenever Q is amenable or second-countable.

Original languageEnglish (US)
Pages (from-to)411-433
Number of pages23
JournalJournal of Operator Theory
Volume46
Issue number2
StatePublished - Sep 1 2001

Keywords

  • C*-algebra
  • Coaction
  • Directed graph
  • Duality
  • Groupoid
  • Skew product

ASJC Scopus subject areas

  • Algebra and Number Theory

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