TY - JOUR

T1 - Applications of non-commutative duality to crossed product c*-algebras determined by an action or co action

AU - Gootman, Elliot C.

AU - Lazar, Aldo J.

N1 - Funding Information:
The work of the first author was partially supported by NSF Grant No. DMS-8601919. A.M.S. (1980) subject classification: 46L55.

PY - 1989/11

Y1 - 1989/11

N2 - For a separable amenable group G and a separable C*-algebra A, let a denote an action of G on A, 6 a coaction of G on A, and GxaA (respectively Gx6A) the corresponding crossed product C*-algebras. We employ non-commutative duality theory to develop a notion of induced representation in the coaction case, and for both actions and coactions, to develop a duality between induction and restriction. We characterize ideals of G invariant under the dual coaction &, as well as ideals of G x6 A invariant under the dual action 6, and show that, in the coaction case, both the notions of induced ideal and of quasi-orbit in the primitive ideal space PR(A) of A are well-defined. For both actions and coactions, the ‘quasi-orbit map’ which maps a primitive ideal of the crossed product algebra to the quasi-orbit in PR(A) ‘over which it lives’ is continuous, open and surjective. As a consequence, if in addition G is compact and A is Type I AF, the crossed product algebra G xa A is also AF.

AB - For a separable amenable group G and a separable C*-algebra A, let a denote an action of G on A, 6 a coaction of G on A, and GxaA (respectively Gx6A) the corresponding crossed product C*-algebras. We employ non-commutative duality theory to develop a notion of induced representation in the coaction case, and for both actions and coactions, to develop a duality between induction and restriction. We characterize ideals of G invariant under the dual coaction &, as well as ideals of G x6 A invariant under the dual action 6, and show that, in the coaction case, both the notions of induced ideal and of quasi-orbit in the primitive ideal space PR(A) of A are well-defined. For both actions and coactions, the ‘quasi-orbit map’ which maps a primitive ideal of the crossed product algebra to the quasi-orbit in PR(A) ‘over which it lives’ is continuous, open and surjective. As a consequence, if in addition G is compact and A is Type I AF, the crossed product algebra G xa A is also AF.

UR - http://www.scopus.com/inward/record.url?scp=0001170933&partnerID=8YFLogxK

U2 - 10.1112/plms/s3-59.3.593

DO - 10.1112/plms/s3-59.3.593

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AN - SCOPUS:0001170933

SN - 0024-6115

VL - s3-59

SP - 593

EP - 624

JO - Proceedings of the London Mathematical Society

JF - Proceedings of the London Mathematical Society

IS - 3

ER -