Duality theory for covariant systems

Landstad, Magnus B.

doi:10.1090/s0002-9947-1979-0522262-6

Cited by 95 publications

(51 citation statements)

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“…Our starting point is recent work [11,13] concerning characterisations of crossedproduct C * -algebras due to Landstad [20] and Quigg [27]. Quigg's theorem identifies crossed products by coactions of G as the systems (A, α) which carry an equivariant embedding j G : (C 0 (G), rt) → (M(A), α).…”

Section: Introductionmentioning

confidence: 99%

Naturality of Rieffel’s Morita Equivalence for Proper Actions

Huef

Kaliszewski

Raeburn

et al. 2009

Algebr Represent Theor

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Abstract. Suppose that a locally compact group G acts freely and properly on the right of a locally compact space T . Rieffel proved that if α is an action of G on a C * -algebra A and there is an equivariant embedding of C 0 (T ) in M (A), then the action α of G on A is proper, and the crossed product A⋊ α,r G is Morita equivalent to a generalised fixed-point algebra Fix(A, α) in M (A) α . We show that the assignment (A, α) → Fix(A, α) extends to a functor Fix on a category of C * -dynamical systems in which the isomorphisms are Morita equivalences, and that Rieffel's Morita equivalence implements a natural isomorphism between a crossed-product functor and Fix. From this, we deduce naturality of Mansfield imprimitivity for crossed products by coactions, improving results of Echterhoff-Kaliszewski-Quigg-Raeburn and Kaliszewski-Quigg-Raeburn, and naturality of a Morita equivalence for graph algebras due to Kumjian and Pask.

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Section: Introductionmentioning

confidence: 99%

Naturality of Rieffel’s Morita Equivalence for Proper Actions

Huef

Kaliszewski

Raeburn

et al. 2009

Algebr Represent Theor

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“…However, since we are concerned in this paper with spectral triples and such a triple involves an explicit Hilbert space, we will work with the reduced crossed product (as was the case in the early work on the subject, e.g. [13,21]). …”

Section: Preliminariesmentioning

confidence: 99%

“…[17,Theorem 13.3.7]) says (among other things) that taking the crossed product of a von Neumann algebra for the action of the modular automorphism group (corresponding to a faithful normal state) transforms a type III factor into a type II ∞ von Neumann algebra. (The C * -algebra version of this is given by the Imai-Takai duality theorem ( [13,Theorem 3.6], [21,Theorem 3], [10,Theorem A.68]) which in its general form, uses the dual coaction -in particular, G does not have to be abelian.) A philosophically similar, but geometrical, situation arose in the work of Connes and Moscovici ( [5]) in the context of diffeomorphism invariant geometry.…”

Section: Introductionmentioning

confidence: 99%

Contractive Spectral Triples for Crossed Products

Paterson¹

2014

MATH. SCAND.

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Abstract. Connes showed that spectral triples encode (noncommutative) metric information. Further, Connes and Moscovici in their metric bundle construction showed that, as with the Takesaki duality theorem, forming a crossed product spectral triple can substantially simplify the structure. In a recent paper, Bellissard, Marcolli and Reihani (among other things) studied in depth metric notions for spectral triples and crossed product spectral triples for Z-actions, with applications in number theory and coding theory. In the work of Connes and Moscovici, crossed products involving groups of diffeomorphisms and even ofétale groupoids are required. With this motivation, the present paper develops part of the Bellissard-Marcolli-Reihani theory for a general discrete group action, and in particular, introduces coaction spectral triples and their associated metric notions. The isometric condition is replaced by the contractive condition.

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“…According to 3.1.8, if G is an abelian group and {31, 7, n} any regular extension of <?M by G, the dual group G acts as automorphisms on 71 in such a way that «/<g))=<&/>Xg) and 7(c3Q = {xe37: a/x)=x for all The following converse is patterned after the results of [18]. We note that if G is discrete, 5.3.1 may be proven more easily using 5.2.4;…”

Section: Dual Actionsmentioning

confidence: 99%

“…[3], [12]) in terms of the existence of an expectation from 57 to I(<3tt). If the group G is abelian we give an alternate characterization, in the spirit of [18], in terms of the existence of an appropriate "dual action" of the dual group of G on 57.…”

Section: Introductionmentioning

confidence: 99%

Cohomology and Extensions of von Neumann Algebras, II

Sutherland¹

1980

Publ. Res. Inst. Math. Sci.

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We develop a theory of extensions of von Neumann algebras by locally compact groups of automorphisms. The emphasis is on the description (from an algebraic point of view) of those extensions of a given von Neumann algebra by a given group which determine a fixed homomorphism from the group into the outer automorphism classes of the given algebra. Thus the study of such homomorphisms occupies a substantial part of the paper; for a large class of examples we are able to determine when such a homomorphism is split, and give a simple algebraic description of the extensions. We then give necessary and sufficient conditions (of an analytic nature) for an extension to be equivalent to a twisted crossed product extension, and give some applications to the study of representations of certain topological groups, and to approximately finite dimensional von Neumann algebras.

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Duality theory for covariant systems

Cited by 95 publications

References 20 publications

Naturality of Rieffel’s Morita Equivalence for Proper Actions

Naturality of Rieffel’s Morita Equivalence for Proper Actions

Contractive Spectral Triples for Crossed Products

Cohomology and Extensions of von Neumann Algebras, II

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