Skip to main content Accessibility help
×
Home

The order dual of an abelian von Neumann algebra

  • Peter G. Dodds (a1)

Extract

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.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      The order dual of an abelian von Neumann algebra
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      The order dual of an abelian von Neumann algebra
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      The order dual of an abelian von Neumann algebra
      Available formats
      ×

Copyright

References

Hide All
[1]Dixmier, J., Les algébres d'opérateurs dans l'espace hilbertien (Gauthier-Villars, Paris, 1957).
[2]Luxemburg, W. A.J. and Zaanen, A. C., Riesz spaces (California Institute of Technology, 1969).
[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.
[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.
[5]Luxemburg, W. A. J., ‘Is every integral normal?’, Bull. Amer. Math. Soc. 73 (1967), 685688.
[6]-Nagy, N. v. Sz., Spektraldarstellung linearer transformation des hilbertschen raumes (Ergebnisse d. Math., Band 5, Springer, Berlin, 1942).
[7]Pallu de la Barrière, R., ‘Sur les algèbres d'opérateurs dans les espaces hilbertiens’, Bull. Soc. Math. Fr., 82 (1954), 151.
[8]Plymen, R. J., ‘C*-algebras and Mackey's axioms’, Comm. Math. Phys. 8 (1968), 132146.
[9]Sakai, S., The theory of W*-algebras (Yale Notes, 1962).
[10]Vulikh, B. Z., Introduction to the theory of partially ordered spaces (Wolters-Noordhoff, Groningen, 1967).
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

The order dual of an abelian von Neumann algebra

  • Peter G. Dodds (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed