Full C^*-Crossed product duality

Abstract

Takai duality for full C*-crossed products holds for twisted actions in the sense of Green and fails for coactions.

On Crossed Products by Coactions and Their Representation Theory

“…and the induced homomorphism 6 X : A/ker <5-» M((yl/ker 6) ® C*(G)) is then an injective coaction of G on A/ker 6. However, as Quigg points out in [25], the splitting of (2.1) does not imply that A is a C*-algebraic direct sum ker 6 ® S(A), as asserted in [27]; rather one recovers A as the C*-algebra…”

Co-universal algebras associated to product systems, and gauge-invariant uniqueness theorems

“…Note that there is an injective Toeplitz representation ψ of X on C * r (F 2 ) determined by ψ(1 p ) := λ F 2 (p). By [30,Example 1.15] or [28,Proposition 2.4], there is a (full) coaction δ F 2 of F 2 on C * r (F 2 ) such that δ F 2 (λ F 2 (g)) = λ F 2 (g) ⊗i F 2 (g) for all g ∈ F 2 . The integrated form of ψ is therefore gauge-compatible.…”

Maximal Coactions

“…If G is any locally compact group, (C * (G), G, δ G ) is maximal, because δ G is the dual coaction on C * (G) = C × id G, and dual coactions are always maximal (Proposition 3.4). The normalization of δ G is the coaction δ n G of G on C * r (G) determined by λ(s) → λ(s) ⊗ u(s) (see [13,Example 2.12]). Thus, if G is nonamenable, δ G is maximal but not normal, and δ n G is normal but not maximal.…”

“…The discussion following [13,Proposition 3.12] shows that the natural coaction of G on C * (G) ⊕ C * r (G) is (when G is nonamenable) neither maximal nor normal. The idea of the proof is as follows: if (A m , G, δ m ) were a maximalization of (A, G, δ), then it would satisfy…”