Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-24T22:33:39.993Z Has data issue: false hasContentIssue false
  • English
  • Français

The order dual of an abelian von Neumann algebra

Published online by Cambridge University Press:  09 April 2009

Peter G. Dodds
Peter G. Dodds
Affiliation:
The Flinders University of South Australia, Bedford Park 5042, South Australia
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The usual technique for dealing with an abelian W*-algebra is to consider it, via the Gelfand theory, as the algebra of all continuous complex-valued functions on an extremally disconnected compact Hausdorff space with a separating family of normal linear functionals. An alternative approach, outlined in [2] and [10], is to develop the theory within the framework of Riesz spaces (linear vector lattices) where the order properties of the self-adjoint operators play an important and natural role. It has been known for a long time that the self-adjoint part of an abelian W*-algebra is a Dedekind complete Riesz space under the natural ordering of self-adjoint operators, but it is only relatively recently that a proof of this fact has been given that is independent of the Gelfand theory, and the interested reader may consult [2] or [10] for the details. This approach is essentially foreshadowed in [6] and provides a very satisfying introduction to the theory of commutative rings of operators. From this point of view, the spectral theorem for self-adjoint operators falls naturally into place as an easy consequence of the spectral theorem of H. Freudenthal. In this paper, the line of approach via Riesz spaces is developed further and several well known results are shown to follow as elementary consequences of the order structure of the algebra.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 18 , Issue 2 , September 1974 , pp. 153 - 160
Copyright
Copyright © Australian Mathematical Society 1974

References

[1]Dixmier, J., Les algébres d'opérateurs dans l'espace hilbertien (Gauthier-Villars, Paris, 1957).Google Scholar
[2]Luxemburg, W. A.J. and Zaanen, A. C., Riesz spaces (California Institute of Technology, 1969).Google Scholar
[3]Luxemburg, W. A. J. and Zaanen, A. C., ‘Notes on Banach function spaces’, Proc. Acad. Sci. Amsterdam; Note VI, A66 (1963), 655–668; Note VII, A66(1963), 669–681; Note VII A67 (1964), 104–119; Note IX, A67 (1964), 360–376; Note X, A67 (1964), 493–506; Note XI, A67 (1964), 507–518; Note XII, A67 (1964), 519–529.Google Scholar
[4]Luxemburg, W. A. J., ‘Notes on Banach function spaces’, Proc. Acad. Sci. Amsterdam; Note XIV, A68 (1965), 230–248; Note XV, A68 (1965), 416–446; Note XVI, A68 (1965), 646–667.Google Scholar
[5]Luxemburg, W. A. J., ‘Is every integral normal?’, Bull. Amer. Math. Soc. 73 (1967), 685–688.CrossRefGoogle Scholar
[6]-Nagy, N. v. Sz., Spektraldarstellung linearer transformation des hilbertschen raumes (Ergebnisse d. Math., Band 5, Springer, Berlin, 1942).CrossRefGoogle Scholar
[7]Pallu de la Barrière, R., ‘Sur les algèbres d'opérateurs dans les espaces hilbertiens’, Bull. Soc. Math. Fr., 82 (1954), 1–51.Google Scholar
[8]Plymen, R. J., ‘C*-algebras and Mackey's axioms’, Comm. Math. Phys. 8 (1968), 132–146.CrossRefGoogle Scholar
[9]Sakai, S., The theory of W*-algebras (Yale Notes, 1962).Google Scholar
[10]Vulikh, B. Z., Introduction to the theory of partially ordered spaces (Wolters-Noordhoff, Groningen, 1967).Google Scholar