Produit Croisé d'une Algèbre de von Neumann par une Algèbre de Kac, II

Enock, Michel; Schwartz, Jean-Marie

doi:10.2977/prims/1195187504

Cited by 32 publications

(30 citation statements)

References 9 publications

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“…Since the unitary 1 W K is an " -cocycle by The¨ore© me IV.3. (ii) of [ES2], the assertion follows from Proposition B.1.…”

Section: On Certain Automorphisms In Eutaaamentioning

confidence: 73%

“…This, together with Corollary B.2, implies that is conjugate to. Since is conjugate to the bidual action of by Proposition IV.5 of [ES2], we conclude that is dual.…”

Section: Proof Withmentioning

confidence: 75%

“…On a " lL 2 9, there is another action of K defined by Ad 1 W K " . It is known [ES2,Proposition IV.5] that the bidual action is conjugate to. The fixed-point algebra a of is defined to be the set fx P a X x x 1 g. Finally, theory of compact Kac algebras almost goes parallel to that of compact groups [ES1,x6].…”

Section: Terminology and Notationmentioning

confidence: 99%

“…We also fix an action of K on a von Neumann algebra a. It is well-known (see [ES2,Proposition II.2]) that E id 9 X a 3 a is a faithful normal conditional expectation from a onto the fixed-point algebra a…”

Section: Dominancy Of Minimal Actionsmentioning

confidence: 99%

“…Let Ã 2 be the canonical injection Ã 2 X p 3 L 2 2. Then, by Section III of [ES2], the operator U 2 on L 2 2 L 2 9 defined by…”

Section: Appendixmentioning

confidence: 99%

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On the dominancy of minimal actions of compact Kac algebras and certain automorphisms in $\mathrm{Aut}(A/A^\alpha)$

Yamanouchi¹

1999

MATH. SCAND.

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We show that any minimal action of a compact Kac algebra with properly infinite fixed-point algebra is dominant. It is also shown that, for such a minimal action, a certain automorphism in eutaaa gives rise to a unitary that lies in the intrinsic group of the dual Kac algebra. A concrete description, in terms of the unitary, of the left co-ideal von Neumann subalgebra determined by the fixed-point algebra of the automorphism is given.

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“…Since the unitary 1 W K is an " -cocycle by The¨ore© me IV.3. (ii) of [ES2], the assertion follows from Proposition B.1.…”

Section: On Certain Automorphisms In Eutaaamentioning

confidence: 73%

“…This, together with Corollary B.2, implies that is conjugate to. Since is conjugate to the bidual action of by Proposition IV.5 of [ES2], we conclude that is dual.…”

Section: Proof Withmentioning

confidence: 75%

Section: Terminology and Notationmentioning

confidence: 99%

Section: Dominancy Of Minimal Actionsmentioning

confidence: 99%

“…Let Ã 2 be the canonical injection Ã 2 X p 3 L 2 2. Then, by Section III of [ES2], the operator U 2 on L 2 2 L 2 9 defined by…”

Section: Appendixmentioning

confidence: 99%

See 3 more Smart Citations

On the dominancy of minimal actions of compact Kac algebras and certain automorphisms in $\mathrm{Aut}(A/A^\alpha)$

Yamanouchi¹

1999

MATH. SCAND.

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Observables in W*‐Algebraic Quantum Mechanics

Amann

1986

Fortschr. Phys.

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The problem of the conceptual and formal characterization of observables in quantum mechanics is of central importance. The very name 'observable' rests on the view that one deals with observable, i.e. measurable quantities.This view gave way to a more subtle analysis, involving a distinction between observables and their measurement. Observables are those quantities which characterize a system intrinsically. They are not measurable in a naive sense. Often their value is accessible but only indirectly and through extended theoretical reasoning. A careful examination of classical mechanics shows that even there say the position of a particle cannot be determined. Measurements never tell if a particle's site has rational or irrational coordinates, they give merely a n interval for it. I n quantum mechanics the matter is still more intricate.Independently of the difficulties associated with measurements, one can define observables with regard to the kinematical group of a quantum system (e.g. the Galilei group), namely as 'suitably' transforming operators under an action of this group. For example, operators for position and momentum can be characterized by their transformation properties under Galilei symmetries in both classical and quantum mechanics without appealing to the correspondence principle. Outside this particular situation one gets into difficulties, since the word 'suitably' has then no specified meaning.

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Observables inW*-Algebraic Quantum Mechanics

Amann¹

1986

Fortschr. Phys.

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Produit Croisé d'une Algèbre de von Neumann par une Algèbre de Kac, II

Cited by 32 publications

References 9 publications

On the dominancy of minimal actions of compact Kac algebras and certain automorphisms in $\mathrm{Aut}(A/A^\alpha)$

On the dominancy of minimal actions of compact Kac algebras and certain automorphisms in $\mathrm{Aut}(A/A^\alpha)$

Observables in W*‐Algebraic Quantum Mechanics

Observables inW*-Algebraic Quantum Mechanics

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