Crossed products of type I af algebras by abelian groups

Applications of Non-Commutative Duality to Crossed Product C* -Algebras Determined by an Action or Coaction

“…The proofs relied essentially on the dual action a of the dual group G of G on the crossed product algebra Gx a A, and on the Takai duality theorem [16], which establishes an isomorphism between the double crossed product algebra G x & (G x a A) and A <8> K(G), K(G) being the algebra of compact operators on L\G). Specifically, the authors obtained a characterization of the G-invariant ideals of G x a A (which, in fact, follows rather easily from Takai duality, or, as in [5, Corollary 2.2] from a representation-theoretic argument involving the dual action a), and also proved that the quasi-orbit map <I>, which assigns to every primitive ideal of G x a A a unique quasi-orbit in the primitive ideal space PR(i4) of A (with respect to the transposed action of G on PR(y4)), is open and surjective, as well as continuous [5,Theorem 2.4]. The latter result depended for its proof on the dual action of G on G x a A, in that the proof proceeded first by showing that for V open with V c PR(G x a A), O(V) = <fc(G • V), and then by utilizing our characterization of the G-invariant ideals of G x a A.…”

“…For a coaction 6 of G on A and a representation n of A, we investigate kernel(Res"(Ind Jt)). Applying this to the coaction a dual to an action, and also using duality theory and the relations established in § 2 between Res, Ind, Res" and Ind", we obtain in Theorem 3.4 our first main goal, an extension of Corollary 2.2 of [5] from abelian to arbitrary amenable G: for an action a of an amenable group G on a C*-algebra A, an ideal I of G x a A is ar-invariant if and only if it is of the form / = G x a J for some unique, cr-invariant ideal J of A. (As pointed out by the referee, this result also follows from the characterization of crossed product algebras given in [9].)…”

“…Then clearly hull/=0~1F, 0(hull/) = 0(0~1F) = F, and all that remains to be shown is the 6-invariance of /. But In [5], it was proved that for an abelian group G acting on a C*-algebra A, the quasi-orbit map of PR(G x a A) into (PR(A)/G) is continuous, open and surjective. The proof involved Takai duality and a result on the structure of ideals of G x a A invariant under the dual action of the dual group G of G on G x a A.…”

“…REMARK. With (G, A, a) as in Theorem 4.11, in order to prove that G x a A is AF, it suffices, by § 3 of [5], to assume that A is liminal with A Hausdorff. Under CROSSED PRODUCT C'-ALGEBRAS 6 2 3 these additional assumptions, the three new facts needed to prove that G x a A is AF are:…”

“…In [5] the authors proved several results concerning the crossed product algebra G x a A formed by the action a of an abelian group G on a C*-algebra. A.…”