Compact group actions on C∗-algebras: An application of non-commutative duality

On Crossed Products by Coactions and Their Representation Theory

“…This gives us a covariant representation (k AII , y' c(G)) oi the reduced system such that j A factors through k All . The integrated form k AI , x j C ( G ) is then a homomorphism Si of the reduced crossed product (AU)x 8…”

Relating properties of crossed products to those of fixed point algebras

“…For pure infiniteness it is Theorem 4.23 of [14], and for strong pure infiniteness it is Proposition 5.11(iii) of [15]. For conditions (7), (8), (9), (10), (11), and (12), use what has been already observed, the fact that stable isomorphism of algebras gives stable isomorphism of their ideals, and fact that stable isomorphism preserves K-theory. Stable isomorphism preserves topological dimension zero because it preserves the primitive ideal space.…”

“…Theorem 3.4 of [10] provides a G-invariant ideal T ⊆ A such that L = C * (G, T, α). If T = A, then, using Lemma 3.1 of [11], find x ∈ A α such that x / ∈ T , so that ϕ(x) / ∈ C * (G, T, α) = L. But ϕ(x) ∈ B ⊆ J ⊆ L by definition. This contradiction shows that T = A, so L = C * (G, A, α).…”

The C*–algebras of Compact Transformation Groups

“…The I FA -A α Morita equivalence has been exploited widely. For example, Gootman and Lazar use non-abelian duality in [19,Theorem 3.2] to prove that for the action of a compact group on A, the crossed product A ⋊ α G is liminal (postliminal) if and only if the fixed-point algebra A α is liminal (postliminal). The "if" direction of this sort of result fails for Fell algebras, as the following example shows.…”